The inclusion-property conjecture for finite point sets in multiprojective spaces

Let $X \subset \mathbb P^{a_1} \times \dots \times \mathbb P^{a_n}$ be a finite set of points with the inclusion property.

Inclusion-property conjecture. $X$ is arithmetically Cohen–Macaulay (ACM).

The conjecture is motivated by the known cases where one factor is $\mathbb P^1$ and where there are only two projective factors. It asks whether the inclusion property implies the ACM property for point sets in products of any number of projective spaces.