Stanley's equidistribution conjecture for alternating permutations and derangements

Let $[n]=\{1,\dots,n\}$ and let $S_n$ be the set of permutations of $[n]$. A permutation $a_1\cdots a_n$ is alternating if $a_1>a_2<a_3>a_4<\dots$, and it is reverse alternating if $a_1<a_2>a_3<a_4>\dots$. Let $d_k(n)$ and $d_k^*(n)$ denote, respectively, the numbers of alternating and reverse alternating permutations in $S_n$ with $k$ fixed points. Let $D_n$ be the number of derangements in $S_n$, that is, permutations with no fixed points.

Stanley's conjecture. For $n\geq4$ and $n\geq5$, respectively,

$$d_{\lceil n/2\rceil}(n)=D_{\lfloor n/2\rfloor},$$

and

$$d^*_{\lceil (n+1)/2\rceil}(n)=D_{\lfloor (n-1)/2\rfloor}.$$

This conjecture asserts an equidistribution between derangements and alternating or reverse alternating permutations having the maximal possible number of fixed points. The source presents it as Stanley's conjecture; no resolution status is supplied here.