The semistar-counting conjecture for integrally closed domains

Let $D$ be an integrally closed domain such that its set of maximal ideals, denoted by $\operatorname{Max}(D)$, is finite. Write $\operatorname{Semistar}(D)$ for the set of semistar operations on $D$.

Semistar-counting conjecture. The domain $D$ is a principal ideal domain (PID) if and only if

$$|\operatorname{Semistar}(D)|$$

is equal to the number of subsets of $2^{\operatorname{Max}(D)}$ that are closed under arbitrary intersections.

This conjecture proposes that, among integrally closed domains with finitely many maximal ideals, the PID property is characterized by the cardinality of the set of semistar operations. The preceding results establish the corresponding count for PIDs and Dedekind domains, while the converse for general integrally closed domains is left open.