The CII maximal-involution conjecture for type B Weyl groups

Let $W=W_{n+1}$ be a type B Weyl group, let $\upsilon_0\in W_n$ be the analogous maximal-length fixed-point-free involution defined in the source, and let

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

The CII maximal-involution conjecture.

$$\hat F^{\mathrm{CII}}_{\upsilon_0}=S_\delta.$$

This conjecture is stated after computational experiments supporting the preceding identities.