Villarreal's determinant conjecture for packing-property clutters

Let $\mathcal C$ be a clutter with clutter ideal generated by monomials $x^{v_1},\dots,x^{v_q}$ of common degree $d\geq 2$. Form the integral matrix

$$B=\left(\begin{array}{ccc} v_1&\cdots&v_q\\ 1&\cdots&1 \end{array}\right)$$

and let $r=\operatorname{rank}(B)$. For an integral matrix, let $\Delta_r(B)$ be the greatest common divisor of its nonzero $r\times r$ subdeterminants.

Villarreal's determinant conjecture. If $\alpha_0(\mathcal C')=\beta_1(\mathcal C')$ for all minors $\mathcal C'$ of $\mathcal C$ and $x^{v_1},\dots,x^{v_q}$ have degree $d\geq 2$, then

$$\Delta_r(B)=1,$$

where $r=\operatorname{rank}(B)$.

The paper calls this a weaker conjecture of Villarreal. It is motivated by the preceding normality conjecture and by a proved determinant-one result under the stronger hypothesis that $\operatorname{gr}_I(R)$ is reduced.