Cohomology-basis conjecture for smooth generalized Hessenberg varieties

Let $w\in S_n$ be smooth, and let $X\in SL(n)$ be regular. For $u\leq w$, let $[\mathcal{Y}_u(X)]\in H^*(\mathcal{Y}_w(X))$ be the corresponding geometric class. Cohomology-basis conjecture. The classes $[\mathcal{Y}_u(X)]$ for $u\leq w$ are linearly independent. If $X$ is semisimple, these classes are fixed by the action of $S_n$. If $X$ is regular unipotent, the classes form a basis of $H^*(\mathcal{Y}_w(X))$. The source presents all three assertions together as conjectural; no resolution is supplied.