Levinson–Ullery conjecture on Cayley–Bacharach sets

Let $Z\subseteq\mathbb{P}^n$ be a Cayley–Bacharach set for $\mathcal{O}(d)$, meaning that every degree-$d$ hypersurface containing all but one point of $Z$ contains all of $Z$. Let $|Z|$ denote the length of $Z$.

Levinson–Ullery conjecture. For $d\geq 1$, if

$$|Z|\leq (e+1)d+1,$$

then $Z$ is contained in a union of positive-dimensional linear subspaces $L_1,\dots,L_k$ of $\mathbb{P}^n$ such that

$$\sum_{i=1}^{k}\dim L_i\leq e.$$

This conjecture gives a structural constraint on small Cayley–Bacharach sets and is presented as a conjecture of Levinson and Ullery. The paper establishes it only under additional hypotheses, so the general statement remains open.