Is this a q-count of Alternating Sign Matrices? The number of Alternating Sign Matrices of size $n$ is well known to be
$\prod_{k=0}^{n-1}\frac{(3k+1)!}{(n+k)!}$. Is it known whether the naive q-analog expression
$$\prod_{k=0}^{n-1}\frac{[3k+1]_q!}{[n+k]_q!}$$
is a polynomial in $q$ with positive coefficients? Does it come from a known statistic on ASM's?
 A: The naive $q$-analog of that expression is naturally a generating function for descending plane partitions by weight (not sure of reference, though), but it doesn't translate to any natural statistic on ASMs/monotone triangles/etc., and it's not known what statistic on ASMs does the trick.
In terms of unimodality, I would imagine there's a way to prove it algebraically by induction. There's a trick along the lines of 'If $f(q)$ is unimodal, and $f(q)/[n]_q$ is a polynomial, then $f(q)/[n]_q$ is also unimodal.' that might work.
A: Perhaps it is more natural to look at Gog or Magog-triangles,
which are special types of GT-patterns, equinumerous with ASMs.
These GT-patterns can then be mapped to SSYTs of triangle shape,
and on these, one can do lots of statistics. 
I tried a few, but no obvious candidate.
EDIT:
This questing has been asked in the end of this article (notices  of the AMS). There seem to be a cyclic sieving phenomenon on ASM,
but they write that they have no idea what the q-statistic is.
However, the CSP might give a lead.
A: It's true these polynomials are not unimodal for n=2 and up, since they all start with coefficient sequence 1 0 1 ... (The reason for this is clear from the definition of descending plane partitions.) For n = 4, 5, 6, they are also non-unimodal in the middle. But the sequence of nonzero coefficients seems to be unimodal for n=7 and above (I've calculated up to n=35). I had observed the non-unimodality of n=4,5,6 several years ago, but had thought it would continue to fail, so it's interesting to me that it seems to be (nearly-)unimodal at n=7 and beyond.
When the ASM is a permutation matrix, the right statistic is a weighted inversion count: weight each inversion pair by the larger number in the pair and then add up all the contributions. (See St000616 on FindStat.) The generating function for this statistic on permutations is an interesting analogue of the q-factorial. (See https://arxiv.org/pdf/1002.3391v2.pdf Corollary 6.) 
I also think looking at the Gog or Magog triangles to try and find the right statistic is a good idea, but it's not one of the usual statistics, as far as I can tell. I've thought about this question quite a lot and would be very interested if anyone is able to make progress on finding this statistic on ASM or TSSCPP.
