In 1922, the Communist Party was firmly in control of the USSR (or at least the Russia part of it), and Vladimir Lenin was the undisputed leader of the Party. Lenin was seriously ill and had to spend time recovering at a residence in Gorki, away from the government in Moscow. While there, he dictated his thoughts to a small circle of confidants that included his wife Nadezhda Krupskaya. Some of those notes started to be distributed in April 1923.
One of those notes is dubbed "Lenin’s Testament". It involves significant comments about the fitness of various party leaders for their respective roles, including in particular a call to remove Stalin from his post of secretary-general. It was dictated in December 1922 - January 1923 with the intention that it would be presented at the April 1923 Party Congress, but was actually kept secret; the document was distributed in May 1924 for that year’s Party Congress. It became known as "Lenin’s testament" due to the timing of publication (shortly after Lenin’s death in January 1924).
The first and obvious question would be: is Lenin’s Testament authentic?. As the next section makes it clear, some scholarly sources think it might not be, but those are in the minority. (I would assume that History Stackexchange is unlikely to dig up much more than what I found, but who knows.)
A second question is: what are the specific arguments of the pro-forgery and anti-forgery camps? My research below does not answer that, because I do not have access to the monograph books, so I am relying on stuff like an internet source citing a book (and even though I assume the quote is authentic, I have no way to know if that was the main argument of the book or a trifling aside).
The third question would be what are the specific documents under question, and what is the timeline (creation, publication) of those specific documents?. As the "context" section says, there are multiple such notes.
Discussion in sources
I was prompted to research and eventually ask this question by listening to the History of Revolutions podcast by Mike Duncan (a general-audience yet decently-researched podcast) (link to the specific episode). Duncan asserts that the testament is more likely than not a forgery by Krupskaya or someone else from their political circle. For the whole context, start listening around 25:32, but the key passage (at 27:46 - 28:30) attributes the following argument to Stephen Kotkin:
(...) I am heavily influenced by the case Stephen Kotkin makes in his biographies of Stalin [Stalin: Paradoxes of Power, 1878–1928] that the provenance of all this miraculous dictation from Lenin is dubious at best. Unlike all the other dictation produced around the same time, the typed-up notes suddenly produced in the spring of 1923 do not have any matching handwritten originals in the archives, nor do they bear Lenin’s initials, which he typically used to mark that yes, this was in fact coming from him. Other dictation from the period has both of these markers of authenticity, but not these later documents that we are here talking about. They were simply typed up and asserted to be Lenin’s words.
More precisely, I believe Kotkin leaves open the possibility that a very ill Lenin did say the words, but only after heavy prompting by Krupskaya, or that Krupskaya invented a significant part of the text rather than all of it - I will call all those possibilities "forgery".
On the other hand, the Wikipedia page (in its current version) describes the testament as authentic, and says Kotkin is the only serious historian to dispute its authenticity:
Historian Stephen Kotkin argued that the evidence for Lenin's authorship of the Testament is weak and suggested that the Testament could have been created by Krupskaya. However, the Testament has been accepted as genuine by other historians, including E. H. Carr, Isaac Deutscher, Dmitri Volkogonov, Vadim Rogovin and Oleg Khlevniuk, and Kotkin's argument was specifically rejected by Richard Pipes.
...and it can be seen from the talk page on Wikipedia that there was a rather lengthy discussion between Wikipedia editors before they agreed on that version.
 is this New Yorker piece, which does not really make any substantive argument (it just says that Kotkin’s argument runs counter to the scholarly consensus).  is Richard Pipes, “The Cleverness of Joseph Stalin,” New York Review of Books, November 20, 2014 (link), to which I do not have access.
 is A review of Stephen Kotkin’s Stalin: Paradoxes of Power, 1878-1928, Fred Williams, June 1st 2015. It does include a (rather lengthy) discussion of Kotkin's argument. Prima facie a post on the "World Socialist Web Site" is probably not the most reliable of sources, however it does includes quotes from Richard Pipes, I assume taken from . Those quotes say (paraphrased) that the Testament was public knowledge by 1926 (published in the New York Times) and that its authenticity had never been questioned at the time (despite Stalin having an obvious interest to do so).
My gut feeling
First of all, I realize that if Kotkin did not convince his colleagues, the most likely reason is that he’s just wrong. As a researcher (albeit not in history), I know that "lone genius overturns current consensus" is a very, very, very rare happenstance, and a much more common occurrence is "respected researcher turns crank".
However, from the sources I discussed above, Duncan/Kotkin seem to me to have the better argument by far. The opposing sources discuss things such as whether Lenin was too ill to dictate, whether Krupskaya had a motive to make an anti-Stalin forgery at the time etc., but do not discuss the material aspect. If the disputed documents do lack features present in similar authentic documents (such as Lenin’s paraph, or an archived handwritten dictation matching the typewritten text), that is a credible dispute to their authenticity.
I do not think the lack of a contemporary dispute about authenticity is all that meaningful either: an early-20th century very high profile forgery took twenty years to be exposed as fraudulent, despite having clear in-text indications (it contained lots of copy-paste of a pre-existing text). A forgery made by Lenin’s closest confident will not have such in-text markers, and the circumstances of creation make it at least plausible.
On the other hand, the timeline seems dubious for a forgery. Duncan/Kotkin questions a whole bunch of documents, some of which were published in April 1923, but the Testament itself was distributed in May 1924. Assuming the Testament was indeed forged around April 1923 or earlier, and that its objective was to discredit Stalin, it should have been published as soon as possible, rather than let Stalin consolidate his hold over the Party over the next year. It would have made little sense to wait any amount of time, and no sense at all to wait past Lenin’s death in 1924 (after which there was no risk that Lenin would publicly denounce the forgery).
Given the above, if I had to bet, I would give it a roughly 10% chance that the Testament is a forgery by Krupskaya. Not much, but not insignificant either.