K-polystability and polar cylinders for higher secant varieties of rational normal curves

Let $C\subseteq \mathbf{P}^d$ be a rational normal curve of degree $d$, and let

$$\Sigma_k:=\Sigma_k(\mathbf{P}^1,\mathscr{O}_{\mathbf{P}^1}(d))\subseteq\mathbf{P}^d$$

be its $k$-th secant variety, defined as the closure of the union of $(k+1)$-secant $k$-planes to $C$. Assume $d\geq 2k+1$. A $(-K_{\Sigma_k})$-polar cylinder is a cylinder in $\Sigma_k$ whose polarization is given by $-K_{\Sigma_k}$. K-polystability and polar-cylinder conjecture. The variety $\Sigma_k$ is K-polystable, and there is a $(-K_{\Sigma_k})$-polar cylinder in $\Sigma_k$. The claim is motivated by the expected matryoshka structure among secant varieties: the paper's results for the first secant varieties suggest analogous behavior for higher secant varieties, but the source gives no resolution evidence for this proposed extension.