The hom- or e-vanishing mutation conjecture for nondegenerate Jacobi-finite QPs

Let $(Q,\mathcal{S})$ be a nondegenerate Jacobi-finite QP. For a pair $(\delta,\epsilon)$ of $\delta$-vectors, call it $\operatorname{hom}$-vanishing when $\operatorname{hom}(\delta,\epsilon)=0$, and call it $\operatorname{e}$-vanishing when ${\rm e}(\delta,\epsilon)=0$. Hom- or e-vanishing mutation conjecture. For any pair $(\delta,\epsilon)$ of $\delta$-vectors, there is a sequence of mutations $\mu_{\boldsymbol{u}}$ such that $(\mu_{\boldsymbol{u}}(\delta),\mu_{\boldsymbol{u}}(\epsilon))$ is either $\operatorname{hom}$-vanishing or ${\rm e}$-vanishing. If true, this would give a mutation procedure leading to one of two vanishing conditions and is presented as a related, more trusted conjecture to the extremal-rank conjecture. The supplied status evidence says that this fact was proved in the proof of Theorem 5.4, so it is recorded as solved.