The freeness conjecture for packing clutters and the associated lattice group

Let $\cal C$ be a clutter with incidence vectors $v_1,\dots,v_q$, and suppose every minor $\cal C'$ satisfies $\tau({\cal C}')=\nu({\cal C}')$. If $x^{v_1},\dots,x^{v_q}$ have degree $d\geq 2$, consider the associated quotient group. Freeness conjecture. The group

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

is free, or equivalently $\Delta_r(B)=1$ where $r={\rm rank}(B)$. The source says that a positive answer to the algebraic Conforti–Cornuéjols conjecture implies this statement; its resolution status is not supplied.