Banaji–Fraser conjecture on characterizing Hausdorff dimension by Hölder stability

Let $n$ be a positive integer, let $\mathcal{K}(\mathbb{R}^n)$ denote the compact subsets of $\mathbb{R}^n$, and let $\dim\colon\mathcal{K}(\mathbb{R}^n)\to[0,n]$ be a dimension. Assume that it agrees with the similarity dimension on each self-similar set satisfying the strong separation condition; is monotone, meaning that $A\subset B$ implies $\dim A\leq\dim B$; is $\sigma$-stable, meaning that

$$\dim\left(\bigcup_{i=1}^{\infty}A_i\right)=\sup_i\dim A_i;$$

is a Borel function; and is Hölder-stable, meaning that for every $\alpha\in(0,1]$ and every $\alpha$-Hölder map $f\colon\mathbb{R}^n\to\mathbb{R}^n$,

$$\dim f(A)\leq\frac{1}{\alpha}\dim A.$$

Banaji–Fraser conjecture. Under these assumptions, $\dim$ is the Hausdorff dimension. This proposes that adding Hölder stability to the standard properties of a dimension characterizes Hausdorff dimension, in contrast with the negative answer obtained for the corresponding question without Hölder stability. The source does not provide a resolution of this modified question.