The DI/DII longest-element conjecture for type D Weyl groups

Let $W=W_n^+$ be the type D Weyl group, let $w_0$ be its longest element, and let

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

For a strict partition $\lambda=(\lambda_1>\dots>\lambda_m>0)$ and a part $a$, write $\lambda\ominus a$ for the partition obtained by removing $a$. The DI/DII longest-element conjecture.

$$\hat F^{\mathrm{DI}}_{w_0}=S_{\delta\ominus a}\quad\text{for }a=\left\lceil\frac n2\right\rceil,$$

and

$$\hat F^{\mathrm{DII}}_{w_0}=S_{\delta\ominus b}\quad\text{for }b=\left\lceil\frac{n+1}{2}\right\rceil.$$

This conjecture lifts enumerative identities predicted by Marberg and Pawlowski and has been tested for $n\leq 7$.