Vanishing minimal volume entropy conjecture for Artin kernels

Let $L$ be a flag complex, let $\mathbb{F}$ be a skew field, and let $\varphi\colon A_L\to\mathbb{Z}$ be an epimorphism. Suppose that $BB_L^\varphi$ is of type $\mathsf{F}$. Minimal-volume-entropy converse conjecture. If

$$\bigoplus_{v\in L^{\mathsf a}}\widetilde{H}_{p-1}(\operatorname{lk}(v);\mathbb{Z})=0,$$

then

$$\omega(BB_L^\varphi)=0.$$

This is proposed as the converse of the preceding positive lower-bound result for minimal volume entropy. The source gives no resolution.