Density-basis differentiation conjecture under maximal-operator finiteness

Let $\mathcal{B}$ be a translation invariant density basis consisting of open sets in $\mathbb{R}^n$, and let $M_{\mathcal{B}}$ be its maximal operator,

$$M_{\mathcal{B}}f(x)=\sup_{x\in R\in\mathcal{B}}\frac{1}{|R|}\int_R|f|.$$

Density-basis differentiation conjecture. If $M_{\mathcal{B}}f(x)<\infty$ almost everywhere, then

$$\lim_{j\to\infty}\frac{1}{|R_j|}\int_{R_j}f=f(x)\quad\text{almost everywhere},$$

where the limit is over an arbitrary sequence of sets $\{R_j\}$ in $\mathcal{B}$ shrinking to $x$.

This proposes a broad generalization of differentiation results for invariant bases. The source gives no evidence that the claim has been resolved.