Bound conjecture for Wilf numbers of semimodules

Let $\Gamma$ be a numerical semigroup and let $W_{\Gamma}(k)$ denote its Wilf function. A $\Gamma$-semimodule is a non-empty, bounded-below subset $\Delta\subseteq\mathbb{Z}$ satisfying $\Delta+\Gamma\subseteq\Delta$; let $W_{\Delta}(2)$ be its Wilf function evaluated at $2$. If $g$ is a gap of $\Gamma$, write $[0,g]$ for the semimodule minimally generated by $0$ and $g$.

Bound conjecture. There exists a semimodule $\Delta$ minimally generated by $[0,g]$ for a gap $g$ of $\Gamma$, such that

$$W_{\Delta}(2)\geq -W_{\Gamma}(e(\Gamma)).$$

This conjecture proposes a lower bound relating the Wilf number of a suitable two-generated semimodule to the Wilf number of the numerical semigroup. The paper introduces it as a conjectural tool toward understanding Wilf's conjecture; no resolution is stated.