Weak $(1,1)$ maximal inequality for ergodic free-group actions

Let $F$ be a finitely generated free group, and let $w\mapsto T_w$ be an ergodic action of $F$ on a probability space $(X,\mathscr B,\mu)$. For $n\geq1$, let $B(\operatorname{id},n)$ be the collection of words in $F$ of length less than $n$. Free-group maximal inequality conjecture. For every $f\in L_1(X)$, one has

$$\left\|\sup_{n\geq1}\frac{1}{|B(\operatorname{id},n)|}\sum_{w\in B(\operatorname{id},n)}|T_wf|\right\|_{L_{1,\infty}(X)}\lesssim\|f\|_{L_1(X)}.$$

The theorem for the infinite regular tree suggests this endpoint ergodic maximal theorem, while the corresponding $L_p$ maximal theorems for $p>1$ were already known; non-amenability prevents the standard transfer argument, so the conjecture remains open.