Nonconstant fixed-point conjecture for the uncentered Hardy–Littlewood maximal operator

Let $X$ be a finite-dimensional Banach space, let $\rho$ be its metric, and let $u$ be a measure on $X$. Write $B(x,r)$ for the ball of radius $r$ centered at $x$, and let $M_\nu$ denote the uncentered Hardy–Littlewood maximal operator on $L^{1}_{\operatorname{loc}}(\nu)$. Assume

$$\{x\in \operatorname{supp}(\nu): r\mapsto \nu(B(x,r))\text{ is discontinuous}\}=\varnothing.$$

A function $f\in L^{1}_{\operatorname{loc}}(\nu)$ satisfying $M_\nu f=f$ is constant.

Nonconstant fixed-point conjecture. Under these assumptions, there is no non-constant fixed point $f\in L^{1}_{\operatorname{loc}}(\nu)$ of $M_\nu$.

The conjecture is motivated by known results for Euclidean spaces and, more generally, measures satisfying related continuity conditions on metric spaces with the Besicovitch covering property. Its status is not resolved in the supplied source.