The generalized Wilf conjecture for Frobenius generalized numerical semigroups

Let $S\subseteq\mathbb{N}^{d}$ be a Frobenius generalized numerical semigroup with Frobenius element $\mathbf{f}=(f^{(1)},\ldots,f^{(d)})$. With $e(S)$ the number of minimal generators and $n(S)$ the number of elements of $S$ lying coordinatewise below some hole, the generalized Wilf conjecture for Frobenius generalized numerical semigroups. One has

$$e(S)n(S)\geq d(f^{(1)}+1)\cdots(f^{(d)}+1).$$

For symmetric Frobenius generalized numerical semigroups, the displayed inequality is proved in the paper; the conjecture proposes it for all Frobenius generalized numerical semigroups.