Sapko's annihilator conjecture for Buchsbaum tangent cones

Let $R$ be the numerical semigroup ring under consideration, let $I$ be its defining ideal, and let $I^*$ be the initial form ideal. Write $\operatorname{gr}_{\mathfrak m}(R)$ for the tangent cone, let $\mathcal M$ be the homogeneous maximal ideal of $\operatorname{gr}_{\mathfrak n}(S)$, and denote the image of $f\in\operatorname{gr}_{\mathfrak n}(S)$ in $\operatorname{gr}_{\mathfrak m}(R)=\operatorname{gr}_{\mathfrak n}(S)/I^*$ by $\bar f$.

Sapko's annihilator conjecture. If $\operatorname{gr}_{\mathfrak m}(R)$ is Buchsbaum, then for some $k\geq 1$,

$$0:_{\operatorname{gr}_{\mathfrak m}(R)}\mathcal M=(\bar{x_3^k})\operatorname{gr}_{\mathfrak m}(R).$$

This is one of Sapko's conjectures concerning tangent cones of 3-generated numerical semigroup rings when the tangent cone is Buchsbaum. Its resolution is not specified in the source.