Symmetry conjecture for crossing distributions of restricted permutations

Let $S_n^k$ denote the set of permutations in $S_n$ having the restriction indexed by $k$, and let ${\rm cr}(\sigma)$ be the number of crossings of a permutation $\sigma$. For an indeterminate $q$, define the crossing-distribution polynomial by

$$\sum_{\sigma \in S_n^k} q^{{\rm cr}(\sigma)}.$$

Symmetry conjecture. For all integers $n$ and $k$ satisfying $1\leq k\leq n$, we have

$$\sum_{\sigma \in S_n^k} q^{{\rm cr}(\sigma)}=\sum_{\sigma \in S_n^{n+1-k}} q^{{\rm cr}(\sigma)}.$$

The paper reports verification by computer up to $n=6$; the general identity is presented as a conjecture and no proof or resolution is supplied.