Herzog-Stamate conjecture on Betti numbers of tangent cones

Let $H = \langle a_1, a_2, \ldots, a_n \rangle$ be a numerical semigroup, and let $I_H^*$ be the defining ideal of the tangent cone of the semigroup ring $K[H]$. Denote by

$$\tilde H = \langle a_1,a_1+1, \ldots,a_n\rangle$$

the numerical semigroup generated by the interval completion of the generators of $H$. For an ideal $I$, write $\beta_i(I)$ for its $i$th Betti number. Herzog-Stamate conjecture. For all $i$,

$$\beta_i(I_H^*) \le \beta_i(I_{\tilde H}^*).$$

The conjecture compares the Betti numbers of the tangent cone of a monomial curve with those of the tangent cone associated with its interval completion. The paper’s abstract and introduction state that this conjecture is established in the paper, so its status is solved.