Strong synchronisation of alternating-group descent and excedance enumerators

Let $\mathcal{A}_n$ be the alternating group in $\mathfrak{S}_n$. Let $A_{n,k}^+$ and $A_{n,k}^-$ count permutations with $k$ descents in $\mathcal{A}_n$ and $\mathfrak{S}_n\setminus\mathcal{A}_n$, respectively, and let $P_{n,k}$ and $Q_{n,k}$ denote the corresponding excedance enumerators. Two sequences are strongly synchronised when each corresponding pair of terms satisfies the synchronisation inequalities defining this relation.

Descent–excedance synchronisation conjecture. For every positive integer $n$, the sequences $(A_{n,k}^+)_{k=0}^{n-1}$ and $(P_{n,k})_{k=0}^{n-1}$ are strongly synchronised. Similarly, the sequences $(A_{n,k}^-)_{k=0}^{n-1}$ and $(Q_{n,k})_{k=0}^{n-1}$ are strongly synchronised.

The claim is motivated by apparent synchronisation despite descents and excedances not being equidistributed on the alternating group. The supplied text gives no resolution.