K-polystability conjecture for regular semisimple Hessenberg varieties and pairs

Let $G$ be a reductive group with Lie algebra $\mathfrak{g}$, let $\mathfrak{g}^{\mathrm{rs}}$ denote its regular semisimple locus, and let $X(s)$ and $Y(s)$ be the associated Fano varieties for $s\in \mathfrak{g}^{\mathrm{rs}}$. Let $\operatorname{\mathbf{Fl}}_n$ be the flag variety and let $c$ satisfy $0<c<1$. K-polystability conjecture. The Fano varieties $X(s)$ and $Y(s)$, as well as the Fano pairs $(\operatorname{\mathbf{Fl}}_n,cX(s))$ with $0<c<1$, are K-polystable for all $s\in \mathfrak{g}^{\mathrm{rs}}$. This conjecture predicts that the natural Fano varieties and pairs arising in the paper admit well-behaved K-moduli, potentially relating their K-moduli compactifications to the GIT compactifications studied here. The source gives no resolution, so the conjecture remains open.