Intersection-cohomology restriction conjecture for generalized Hessenberg varieties

Let $G$ be the group, let $X\in G$ be regular, let $w\in W$, and let $\mathcal{Y}_w(X)$ be the corresponding variety. Let $\Omega_w$ be the Schubert variety and let $i\colon\mathcal{Y}_w(X)\to\Omega_w$ be the natural map defined by

$$i(gB)=(XgB,gB).$$

Intersection-cohomology restriction conjecture. For $X\in G$ regular and $w\in W$,

$$IC_{\mathcal{Y}_w(X)}=i^!(IC_{\Omega_w}).$$

The source motivates this by the expected relationship between the intersection cohomology of generalized Hessenberg varieties and Schubert varieties; it gives no resolution.