The converse conjecture for ACM point sets and the condition $N_1(X) \in \mathcal D(X)$

Let $X=A_X\cup B_X$ be a set of points satisfying the assumptions stated in the surrounding theorem, and let $N_1(X)$ and $\mathcal D(X)$ denote the associated numerical invariant and distinguished set. Assume that $X$ is arithmetically Cohen–Macaulay (ACM).

Converse conjecture. Then

$$N_1(X)\in\mathcal D(X).$$

This is proposed as the converse of the preceding theorem, which establishes the implication in the other direction under the same assumptions. The source gives no resolution of the converse.