Bottom-heaviness conjecture for q-Stirling coefficients

For a finite coefficient sequence $(a_0,\ldots,a_n)$, call it bottom heavy when $a_k\geq a_{n-k}$ for $k<n/2$. Let $S[n,k]$, $c[n,k]$, $S_B[n,k]$, and $c_B[n,k]$ be the type $A$ and type $B$ $q$-Stirling polynomials. Bottom-heaviness conjecture. For each $n,k$, the coefficient sequences of all four polynomials are bottom heavy. The claim describes a common asymmetry observed in the Stirling polynomials; the source gives no resolution status.