Mustăța's Minimal Resolution Conjecture for points on projective varieties

Let $X\subset\mathbb P^n$ be a projective variety of dimension $d\geq 1$, with $\operatorname{reg}(X)=m$ and Hilbert polynomial $P_X$. Let $\delta$ satisfy $P_X(r-1)\leq\delta<P_X(r)$ for some $r\geq m+1$, and let $\Gamma$ be a set of $\delta$ points on $X$ in general position. If

$$0\to F_n\to F_{n-1}\to\cdots\to F_1\to R\to R/I(X)\to0$$

is a minimal free $R$-resolution, then $R/I(\Gamma)$ has a minimal free resolution of the displayed type, with $a_{r+i}^i a_{r+i}^{i+1}=0$ for $i=1,\ldots,n-1$. Mustăța's Minimal Resolution Conjecture. The minimal free resolution of $R/I(\Gamma)$ is obtained by adjoining to each $F_i$ precisely the two indicated consecutive-degree summands, subject to

$$a_{r+i}^i a_{r+i}^{i+1}=0\qquad (i=1,\ldots,n-1).$$

This generalizes the Minimal Resolution Conjecture for points in general position in projective space; the source notes partial results and counterexamples in related settings, while the stated variety version is presented as a conjecture.