Component correspondence for Grassmann cactus varieties and Hilbert schemes

Let $X$ be a projective variety and let $r,k$ be positive integers. For a “sufficiently ample” embedding $X\subset {\mathbb P}V$, consider the Grassmann cactus variety $\mathfrak{K}_{r,k}(X)$ and the Hilbert scheme $\operatorname{Hilb}_r(X)$ of degree-$r$ zero-dimensional subschemes of $X$. A general element of a Hilbert-scheme component has socle dimension at most $k$ when its general member does.

Component correspondence conjecture. The irreducible components of $\mathfrak{K}_{r,k}(X)$ are in bijection with the irreducible components of $\operatorname{Hilb}_r(X)$ whose general element has socle dimension at most $k$.

The paper presents this as an effectiveness question and states that precise answers are not yet available; the required meaning of “sufficiently ample” is not specified in the supplied text.