The Generalized Wilf Conjecture for generalized numerical monoids

Let $n\geq 2$, and let $T$ be a generalized numerical monoid (GMS), meaning a complement-finite submonoid of $\mathbb N^{n-1}$. Let $H(T)=\mathbb N^{n-1}\setminus T$ be its hole set. Define

$$c(T)=\left|\{\mathbf a\in\mathbb N^{n-1}:\mathbf a\leq\mathbf b\text{ for some }\mathbf b\in H(T)\}\right|$$

and

$$n(T)=\left|\{\mathbf a\in T:\mathbf a\leq\mathbf b\text{ for some }\mathbf b\in H(T)\}\right|,$$

where $\mathbf a\leq\mathbf b$ means coordinatewise inequality, and let $e(T)$ be the cardinality of a minimal generating set of $T$. Generalized Wilf Conjecture. One has

$$(n-1)c(T)\leq e(T)n(T).$$

This conjecture is the commutative generalized numerical-monoid analogue of the Unipotent Wilf Conjecture and is attributed in the paper to Cisto, Delgado, Farrán, Failla, Pérez, and Utano. Its general status is not resolved here.