Bound conjecture for the minimum Wilf number of a gap

Let $\Gamma$ be a numerical semigroup, let $W_{\Gamma}(e(\Gamma))$ be its Wilf number, and for each gap $g$ let $W(g)$ denote the Wilf number associated with that gap. Then

Bound conjecture. For any semigroup $\Gamma$,

$$\min(W(g))\geq -W_{\Gamma}(e(\Gamma)).$$

This conjecture proposes a lower bound for the Wilf number of a gap and is motivated by numerical examples and the study of Wilf functions. The paper presents it as open and as a possible route toward Wilf's conjecture.