Strong differentiation conjecture for axis-parallel rectangles

Let $f$ be a function on $\mathbb{R}^2$, let $\mathcal{B}$ be a collection of open sets, and define

$$\mathcal{B}_2=\{R\in\mathcal{B}:R\text{ is a rectangle in }\mathbb{R}^2\text{ whose sides are parallel to the coordinate axes}\}.$$

Define

$$\widetilde{M}_{\mathcal{B}}f(x)=\lim_{r\to 0}\sup_{x\in R\in\mathcal{B},\,\operatorname{diam}R<r}\frac{1}{|R|}\int_R|f|.$$

Strong differentiation conjecture. If $\widetilde{M}_{\mathcal{B}}f(x)$ is finite almost everywhere, then $f$ is strongly differentiable.

This would strengthen the theorem of Besicovitch by replacing the full strong maximal operator with its small-scale version. The source presents it as a future direction and gives no resolution.