K-polystability conjecture for rank-2 Fano threefolds of degree 28

Let $X$ be the smooth Fano threefold obtained by blowing up a smooth quadric threefold $Q\subset\mathbb{P}^4$ along the smooth twisted rational quartic curve $C_4$, so that $\operatorname{Pic}(X)\cong\mathbb{Z}^2$ and $(-K_X)^3=28$. The quadric $Q$ is equipped with the induced $\mathrm{SL}_2(\mathbb{C})$-action. K-polystability conjecture. The smooth Fano threefold $X$ is K-polystable if and only if the quadric $Q$ is GIT-polystable with respect to the $\mathrm{SL}_2(\mathbb{C})$-action. The conjecture proposes a precise correspondence between K-polystability of these Fano threefolds and GIT-polystability of their defining quadrics; the supplied source does not indicate that it has been resolved.