Rational smoothness conjecture for regular generalized Hessenberg varieties

Let $X$ be an element of the group under consideration, let $\underline{s}$ be a reduced word, and let $\mathcal{Y}_{\underline{s}}(X)$ be the associated variety. A variety is rationally smooth when its intersection cohomology complex agrees with the shifted constant sheaf, namely $IC_{\mathcal X}=\mathbb C_{\mathcal X}[\dim(\mathcal X)]$. Rational smoothness conjecture. When $X$ is regular, the varieties

$$\mathcal{Y}_{\underline{s}}(X)$$

are rationally smooth. The regular semisimple case is noted to be smooth, but the general regular case remains conjectural in the source.