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](http://www.findstat.org/StatisticsDatabase/St000616/) on [FindStat](http://www.findstat.org).) 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.