Interval-completion bound for the defining ideal of a numerical semigroup tangent cone

Let $H=\langle a_1,\dots,a_r\rangle$ be a numerical semigroup, and let $\widetilde H$ be its interval completion, the numerical semigroup generated by all integers in the interval $[a_1,a_r]$. Let $I_H^*$ and $I_{\widetilde H}^*$ denote the defining ideals of the tangent cones of $H$ and $\widetilde H$, respectively. Interval-completion bound conjecture. For every numerical semigroup $H$,

$$\mu(I_H^*)\leq\mu(I_{\widetilde H}^*).$$

The conjecture compares the number of minimal generators of the tangent-cone defining ideal with that of the interval completion, whose generators fill the interval between the smallest and largest generators of $H$. The source gives no proof or evidence of resolution, so the status remains open.