Villarreal's torsion-free quotient conjecture for uniform clutters

Let $A$ be a $\{0,1\}$-matrix whose columns each contain the same number $d\geq 2$ of $1$'s, with column vectors $v_1,\ldots,v_q$. Let $\mathcal{C}$ be the clutter associated with $A$, and let $\mathcal{C}'$ range over all minors of $\mathcal{C}$. For each minor, let $\alpha_0(\mathcal{C}')$ be the smallest cardinality of a minimal vertex cover and let $\beta_1(\mathcal{C}')$ be the maximum number of pairwise independent edges. Villarreal's conjecture. If

$$\alpha_0(\mathcal{C}')=\beta_1(\mathcal{C}')$$

for every minor $\mathcal{C}'$ of $\mathcal{C}$, then

$$\mathbb{Z}^{n+1}/\big((v_1,1),\ldots,(v_q,1)\big)$$

is torsion-free. The source describes this as a weaker conjecture following a proposition that reducedness of the associated graded ring forces the displayed quotient group to be torsion-free. Its resolution status is not given in the supplied text.