The product-of-projective-spaces conjecture for Kazhdan–Lusztig polynomials

Let $k+2l=n$. Define $u_{l}^{k}\in\mathrm{S}_{n}$ by

$$u_{l}^{k}=(n-l+1,\ldots,n,l+1,\ldots,n-l,1,\ldots,l),$$

which moves the first $l$ strings to the last $l$ positions, fixes the middle $k$ strings, and moves the last $l$ strings to the first $l$ positions. Let $w_{0}^{l}$ be the longest element of $\mathrm{S}_{l}$ and set

$$w_{l}^{k}=(w_{0}^{l}\otimes\operatorname{id}_{k}\otimes w_{0}^{l})\circ u_{l}^{k}.$$

Product-of-projective-spaces conjecture. The Kazhdan–Lusztig polynomial indexed by $w_{l}^{k}$ is the Hilbert–Poincaré polynomial of $(\mathbb{C}\mathbb{P}^{l})^{\times(k-1)}$:

$$\mathrm{P}_{w_{l}^{k}}=(1+\mathtt{v}+\cdots+\mathtt{v}^{l})^{k-1}.$$

The proposed formula gives an explicit family of Kazhdan–Lusztig polynomials and is presented as a more specific conjecture motivated by the preceding growth discussion; no proof or resolution is supplied.