The Mayans used a 365 day year so is much less accurate than even the Julian calendar. The idea that it is highly accurate is based on a claim that they added 12.5 "leap days" all in one go every 52 years - see answer to this question.
For the Mayans anyway this would need good solid evidence to confirm it. It seems inconsistent with their system of the "long round".
52 long count years (360 days) are one Tzolkin (260 days) short of 52 Haab years (365 days, same as 72 Tzolkins)
So every 52 years the long count will move back one Tzolkin. After 72 of those then the long count will move back one calendar round and so will coincide again (after 73 * 52 long count years).
After 52 * 72 years or 3744 of the Haab years, or 73 * 52 = 3796 long count years, the whole cycle would repeat, you get the same long count, Haab and Tzolkin date as you got 3744 Haab years previously
They were aware of this cycle at least according to a preprint
Chanier, T., 2013. The Mayan Long Count Calendar. arXiv preprint arXiv:1312.1456..
"In particular, on page 24 of the Dresden codex is written the so-called Long Round number noted 22.214.171.124.0 or 1366560 days, a whole multiple of the Tzolk’in, the Haab’, the Tun, Venus and Mars synodic periods, the Calendar Round and the Xultun number X0:LR= 1366560 = 5256×260 = 3744×365= 3796×360 = 2340×584 = 1752×780 = 72×18980= 4×341640.."
But how could they have the Long Round 126.96.36.199.0 for all three cycles to repeat, if they were inserting 12.5 "leap days" all in one go every 52 Haab years?
[This is a rhetorical question, it doesn't make sense that they would have the long round system if they were inserting 12.5 "leap days"]
Let's look at the effect of inserting those 12.5 days.
Then after 72 of the calendar rounds you would need to insert 72 *12.5 = 900 days.
Then you have 1366560 + 900 = 1367460 days in 72 of the Haab and Tzoltin cycles and 1366560 days in the 73 * 52 long count years. and they wouldn't match.
With those extra 12.5 days then two calendar rounds gives you 104*365+25 = 37985 days, then for the long count years to coincide you need a least common multiple of that with 360, which turns out to be 2,734,920 (using online lcm calculator).
It is 144 calendar rounds this time and 7597 long count years which would be notated as 188.8.131.52.0 instead of 184.108.40.206.0