Parity log-concavity conjecture for type B q-Stirling coefficients

For each pair of nonnegative integers $n$ and $k$, let $S_B[n,k]$ and $c_B[n,k]$ be the type $B$ $q$-Stirling polynomials. A coefficient sequence is parity log-concave when its even-indexed and odd-indexed subsequences are each log-concave. Parity log-concavity conjecture. For each $n,k$, the coefficients of $S_B[n,k]$ and $c_B[n,k]$ are positive and parity log-concave, hence parity unimodal. The claim is motivated by computational evidence after examples showing that ordinary unimodality can fail in type $B$; the source gives no resolution status.