Herzog–Stamate bound for generators of monomial curve ideals

Let $\Gamma$ be a numerical semigroup, let $I_\Gamma$ be its defining ideal, and let $\mathrm{wd}(\Gamma)$ denote its width. The number $\mu(I_\Gamma)$ of generators of $I_\Gamma$ satisfies

Herzog–Stamate conjecture.

$$\mu(I_\Gamma) \leq {\mathrm{wd}(\Gamma)+1 \choose 2}.$$

This conjecture gives an explicit upper bound for the first Betti number of a monomial curve in terms of its width. It is a consequence of a stronger conjecture concerning the defining ideal of the tangent cone, and no progress had been made on it since it was stated.