Last year, a team headed by Steve Fine of the Yeshiva University Center for Israeli Studies in New York which had been examining portions of the arch since 2012 announced:
High resolution three-dimensional scans of the Menorah and the
deification reliefs were made, and part of the Menorah relief was
examined to determine whether any traces of paint decoration were
preserved. A Breuckmann GmbH 3D scanner was used for the data
capture. UV-VIS spectrometry was employed to detect color on the
marble reliefs.
Traces of yellow ochre were found on the arms and base of the Menorah.
There's a short video, ArchOfTitus SpoilsPanel 040717 (4m22secs), which the reconstruction team says "chronicles our methods and progress." According to the article The Arch of Titus’s Menorah Panel in Color
These results aligned with the Jewish historian Josephus’s account of
the Roman victory parade, wherein he describes the menorah as being
gold.
The team then added color to the rest of the panel—bringing the
ancient scene to life. They colored the background sky blue, the
tunics off-white, the overgarments reddish-purple, the wreaths green,
the laurel berries purple, the sacred vessels gold, the trumpets
silver, and the leather and wood brown. They colored the arch (in the
far right of the panel) white, black and gold. Further, they added
labels to the three signs held by the Roman victors; these labels were
based loosely on Josephus’s text.
The result can be seen below:
Source: Coloring in the troubled history of a renowned Roman arch
Of course, this is just one part of the arch and as Peter J. Schertz, one of the team members, admits, it is
a hypothetical and extremely speculative reconstruction
However, as Fine explains in a recent article, it wasn't pure guesswork though (my highlighting):
The intensity of the pigments as reconstructed is based upon the color
value of the yellow ochre of the menorah, with none of the nuance that
would have come with final finishing and the the fading that comes with
exposure to the sun... We colored the sky blue following the most
common color for such things in wall paintings from Pompeii and
Herculaneum. Since military tunics could be wool or linen, and are
shown that way on the wall paintings, we decided on a shade of
off-white - except for the overgarments, which we made a shade of
reddish-purple worn by people of high status... The wreaths worn by
the celebrants...are green, the color of the laurel leaves that
compose them, while the laurel berries are purplish.
We colored the skin and hair in Mediterranean shades, and the leather
and wood in shades of brown. Pillows are shown that support the heavy
menorah and the table...We colored them a slightly darker shade to contrast
with the linen. The signs, tabulae ansatae, literally "horned tablets," are set in frames, which we colored bronze in contrast with the gold of the sacred vessels.
On restoration, this was probably done for as long as the state could afford it or there were wealthy Romans looking to improve their reputation by paying for it. By the 4th century AD, though, the city and the economy were in serious decline. Wealthy Roman families either died out or moved to Constantinople, while the population declined from over a million during the 2nd century AD to around 30,000 in 550 AD. All this meant less city income from taxation and, inevitably, things began to fall apart as buildings were abandoned and money ran out.
If restoration work had been done in previous centuries, it most likely ceased in the 4th century or perhaps the early 5th century, but we don't really know. It is unlikely that it ceased because of Titus' reputation; as Wikipedia notes,
Titus's record among ancient historians stands as one of the most
exemplary of any emperor.
It is possible, though, that as Christianity spread in the late western empire, Titus' reputation suffered, but it was the early 19th century Pope Pius VII who initiated restoration work at that time.
Although further research is ongoing, the ravages of time, the Arch's use as part of a medieval fortification and the 19th centruy restoration work have all contributed to some of the orginal stonework being lost, thus making it less likely that more colours will be detected by modern science.
Other sources
Kevin Twine, The City in Decline: Rome in Late Antiquity (pdf)
Vinzenz Brinkmann, Oliver Primavesi, Max Hollein (eds), The Polychromy of Antique and Medieval Sculpture
Laura Eiford, The Arch of Titus
