The plane configuration conjecture for Cayley–Bacharach point sets

Let $\Gamma \subset \mathbb{P}^n$ be a finite set of points satisfying the Cayley–Bacharach condition $\mathrm{CB}(r)$: every degree-$r$ hypersurface containing all but one point of $\Gamma$ contains the remaining point. A plane configuration is a union $\mathcal{P}=P_1\cup\cdots\cup P_k\subseteq\mathbb{P}^n$ of positive-dimensional linear spaces, with dimension

$$\dim(\mathcal{P})=\sum_i\dim(P_i).$$

Plane configuration conjecture. If

$$|\Gamma|\leq (d+1)r+1,$$

then $\Gamma$ lies on a plane configuration $\mathcal{P}$ of dimension $d$. The conjecture seeks a general geometric description of finite Cayley–Bacharach sets beyond the known case where the corresponding cardinality bound forces the points to lie on a line; the intermediate case yields configurations such as a plane conic or two skew lines. Its validity in the stated generality is not established in the supplied source.