Cisto's Generalized Wilf conjecture for N^p-semigroups

Let $S$ be an $\mathbb{N}^p$-semigroup, let $\mathcal{H}(S)$ denote its set of holes, and write $x\leq_{\mathbb{N}^p}h$ for the coordinatewise order. Define

$$\nu(S)=\sharp\{x\in S\mid x\leq_{\mathbb{N}^p}h\text{ for some }h\in\mathcal{H}(S)\},$$ $$\gamma(S)=\sharp\{x\in\mathbb{N}^p\mid x\leq_{\mathbb{N}^p}h\text{ for some }h\in\mathcal{H}(S)\}.$$

Cisto's Generalized Wilf conjecture. The inequality

$$\nu(S)e(S)\geq p\,\gamma(S)$$

should hold.

This generalizes Wilf-type inequalities to $\mathbb{N}^p$-semigroups; the source gives no resolution status.