The maximal fixed-point-free involution conjecture for type D Weyl groups

Let $W=W_n^+$ be the type D Weyl group. For $n\geq2$, let $\upsilon_0^+\in W_n^+$ be a maximal-length element of the relevant fixed-point-free involution set, as defined in the source. Let

$$\delta=(n,n-1,\dots,2,1).$$

The maximal fixed-point-free involution conjecture.

$$\hat F^{\mathrm{DI}}_{\upsilon_0^+}=S_\delta\quad\text{when }n\text{ is even},$$

and

$$\hat F^{\mathrm{DII}}_{\upsilon_0^+}=S_\delta\quad\text{when }n\text{ is odd}.$$

The conjecture has been tested for $n\leq7$; the element $\upsilon_0^+$ is non-unique in general.