Weighted strong Poincaré inequality for biparametric rectangles

Let $w$ be a weight in $A_{p,\mathfrak{R}}$ for $p>1$. For a rectangle $R=I_1\times I_2$, let $\pi_R(f)$ denote the projection used in the Poincaré inequality, and let $\nabla_x\nabla_y f$ be the mixed gradient. Weighted strong Poincaré conjecture. For every $f\in C^2(R)$,

$$\|f-\pi_R(f)\|_{L^{p}(R,w)}\lesssim [w]_{A_{p, \mathfrak{R}}}^{\frac{1}{p-1}+\frac{1}{p}}\ell(I_1)\ell(I_2)\,\|\nabla_x\nabla_y f\|_{L^{p}(R,w)}.$$

The preceding theorem establishes only the corresponding local weak-type estimate, because the maximal operator is not bounded on $L^1$. In the classical cube setting, truncation can sometimes upgrade weak estimates to strong ones, but whether this works for mixed gradients in the present biparametric rectangle geometry is left as a conjecture.