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?

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?