Width bound and equality characterization for the defining ideal of a numerical semigroup tangent cone

Let $H$ be a numerical semigroup, let $I_H^*$ denote the defining ideal of its tangent cone, let $\mu(H)$ be the number of minimal generators of $H$, and let $\width(H)$ be the difference between the largest and smallest generators in a minimal generating set of $H$. Width bound conjecture.

$$\mu(I_H^*) \leq {\width(H)+1 \choose 2}.$$

If $\mu(H) \geq 2$, equality holds if and only if there exist integers $w,k \geq 1$ such that

$$H=\langle kw+1,kw+2,\dots,(k+1)w+1\rangle.$$

This conjecture would give an explicit bound in terms of the width for the number of minimal generators of the tangent-cone defining ideal, refining the uniform boundedness established in the preceding corollary. The source reports that computer calculations suggest the statement; no proof or resolution is provided here.