Sawyer-type weak inequality for the multilinear maximal operator with rectangular weights

Let $1\leq p_1,\dots,p_m<\infty$ and define $p$ by

$$\frac{1}{p}=\frac{1}{p_1}+\dots+\frac{1}{p_m}.$$

Let $w_1,\dots,w_m$ and $\nu$ be weights such that $(w_1,\dots,w_m,\nu)\in A_{\vec P}^{\mathcal R}$, and let $v$ be a weight satisfying $\nu v^p\in A_{\infty}$. Sawyer-type conjecture. There exists a constant $C>0$ such that

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

for every vector of measurable functions $\vec f=(f_1,\dots,f_m)$. This conjecture proposes the complete multilinear analogue of Sawyer-type inequalities for the maximal operator under $A_{\vec P}^{\mathcal R}$ weight conditions; its resolution is not specified in the supplied text.