Strong synchronisation of alternating-group descent enumerators

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

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

This conjecture would imply log-concavity of both descent enumerator sequences. The surrounding discussion also relates it to conjectures on real-rootedness of the alternating-group descent polynomials, but no resolution is supplied here.