Unified weighted-condition conjecture for the multilinear maximal operator

Let $m$ be a positive integer, let $\vec{w}=(w_1,\ldots,w_m)\in A_{\vec{1}}$, let $v$ be a weight with $v^{1/m}\in A_{\infty}$, and define

$$\nu=w_1^{1/m}\cdots w_m^{1/m}.$$

Unified weighted-condition conjecture. There is a constant $C$ such that

$$\left\|\frac{\mathcal{M}(\vec f\,)(x)}{v}\right\|_{L^{1/m,\infty}(\nu v^{1/m})}\leq C\prod_{i=1}^m\|f_i\|_{L^1(w_i)}.$$

This would unify the two sufficient conditions already obtained: $\vec w\in A_{\vec 1}$ with $\nu v^{1/m}\in A_\infty$, and $w_i\in A_1$ for every $i$ with $v\in A_\infty$. The supplied status evidence says that this fact was proved in Remark 7.5 of the cited work, so the conjecture is solved.