Herzog–Srinivasan conjectures on generators of initial ideals of numerical semigroups

Let $H$ be a numerical semigroup minimally generated by $a_1<\dots<a_n$. Let $\operatorname{width}(H)=a_n-a_1$, let $I_H^*$ denote the ideal of initial forms associated to the defining ideal of the semigroup ring $K[H]$, and let $\mu(J)$ denote the minimal number of generators of an ideal $J$. Let $\widetilde H$ be the semigroup generated by all integers in the interval $[a_1,a_n]$. Herzog–Srinivasan's generator conjectures. For every numerical semigroup $H$,

$$\text{(i)}\quad \mu(I_H^*)\leq {\operatorname{width}(H)+1\choose 2},$$

and

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

If true, the second assertion would imply the first, and the interval completion's arithmetic-sequence structure would yield effective bounds. The source gives no resolution status beyond identifying these as conjectures.