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?