Involution word tableau enumeration conjecture for type D

Let $D_n$ be the finite Coxeter group of signed permutations with an even number of negative letters, with longest element $w_n^D$. For the identity automorphism, let $\underline{\hat{\mathcal{R}}}(w_n^D)$ denote the involution words with each occurrence of $t_1'$ replaced by $t_1$; for the automorphism $*$ interchanging $t_1$ and $t_1'$ and fixing $t_i$ for $i\in[2,n-1]$, let $\underline{\hat{\mathcal{R}}}_*(w_n^D)$ be defined analogously. Write

$$\lambda=(n-1,n-2,\ldots,\lfloor n/2\rfloor,\lfloor n/2\rfloor,\ldots,2,1)$$

and

$$\mu=(n-1,n-2,\ldots,\lceil n/2\rceil-1,\lceil n/2\rceil-1,\ldots,2,1).$$

Involution word tableau enumeration conjecture. If $n\geq 3$, then

$$\left|\underline{\hat{\mathcal{R}}}(w_n^D)\right|=|\operatorname{SYT}(\lambda)|$$

and

$$\left|\underline{\hat{\mathcal{R}}}_*(w_n^D)\right|=|\operatorname{SYT}(\mu)|.$$

These are proposed analogues for twisted involution words of the known type D reduced-word enumeration by standard Young tableaux. The claim is presented as an apparent analogue and no resolution is supplied in the source.