Acyclicity conjecture for Gröbner-smoothable simplicial complexes

Let $K$ be a field, and let $\Gamma$ be a $d$-dimensional simplicial complex. Say that $\Gamma$ is Gröbner smoothable over $K$ if there exists a homogeneous ideal $J\subseteq S$ and a monomial order $<$ such that

$$\operatorname{in}_<(J)=I_\Gamma$$

and the projective variety $\mathbb{V}(J)\subseteq\mathbb{P}^n$ is connected and smooth over $K$.

Acyclicity conjecture. If $\Gamma$ is Gröbner smoothable over $K$, then $\Gamma$ is acyclic over $K$.

This is an equivalent formulation of the cohomological Herzog conjecture used in the paper's proof of the curve case. The source does not state that this formulation has been resolved in general, so its status beyond the proved curve case is open.