Integral-group and Ehrhart characterization conjecture for uniform clutters

Let $\mathcal C$ be a uniform clutter with edge incidence vectors $v_1,\ldots,v_q$, and suppose its packing property holds. Define

$$B=((v_1,1),\ldots,(v_q,1))$$

as the matrix whose columns are the augmented vectors, let $r$ be the rank of $B$, let $\Delta_r(B)$ be the corresponding determinantal divisor, and set

$$P=\operatorname{conv}(v_1,\ldots,v_q).$$

Here $\overline{K[Ft]}$ is the integral closure of the homogeneous ring $K[Ft]$, and $A(P)$ is the Ehrhart ring of $P$. Integral-group and Ehrhart characterization conjecture. If $\mathcal C$ is a uniform clutter with the packing property, then all of the following equivalent conditions hold: (a) $\mathbb{Z}^{n+1}/((v_1,1),\ldots,(v_q,1))$ is a free group; (b) $\Delta_r(B)=1$; (c) $B$ diagonalizes over $\mathbb{Z}$ to an identity matrix; and (d) $\overline{K[Ft]}=A(P)$. The source presents this as a consequence suggested by the preceding conjecture, but gives no resolution status.