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)][1]. 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.



  [1]: http://www.ams.org/notices/201402/rnoti-p169.pdf