The conjecture on uniform continuity and weakly Euclidean points

Let $(X,d,m)$ be the metric measure space under consideration, let $k$ be the dimension parameter, and define

$$\mathcal{M}_k:=\left\{x\in X\;;\;\liminf_{r\rightarrow0}\frac{m(B_r(x))}{r^k}=0\right\}.$$

Assume that the function $r\mapsto m(B_r(x))/r^k$ is uniformly continuous for $r\geq 0$. Let $\underline{\mathcal{WE}}_{k}$ denote the set of weakly Euclidean points of dimension $k$.

The conjecture on uniform continuity and weakly Euclidean points. Under these assumptions,

$$\underline{\mathcal{WE}}_{k}\subset \mathcal{M}_k.$$

This generalizes the preceding closedness argument for $\mathcal{M}_1$ and relates the local volume growth of weakly Euclidean points to the vanishing lower $k$-dimensional density. The source does not provide a resolution of this conjecture.