Sidman–Smith conjecture on determinantal ideals of cactus varieties

Let $X$ be a smooth projective variety of dimension $n$. Let $\kappa_k(X,L)$ be the $k$-th cactus variety, namely the closure of the union of the $k$-planes spanned by finite subschemes of $X$ of length $k+1$, and let $L=A\otimes B$ with $A$ and $B$ sufficiently ample. The catalecticant matrix $\operatorname{Cat}(A,B)$ is the matrix of the multiplication map $H^0(X,A)\otimes H^0(X,B)\to H^0(X,L)$. Sidman–Smith conjecture. The ideal $I(\kappa_k(X,L))$ is generated by the $(k+2)\times(k+2)$-minors of $\operatorname{Cat}(A,B)$; in particular, $I(\kappa_k(X,L))$ is determinantally presented whenever $L$ is sufficiently ample. This generalizes the Eisenbud–Koh–Stillman conjecture by replacing secant varieties with cactus varieties, which is necessary because the catalecticant minors may vanish on the cactus variety as well as on the secant variety. The source does not state a resolution of this conjecture.