Fractal dimension conjecture for optimal branched-transport shapes

Let $A$ be an optimal unit-volume set for irrigation from a single source at the origin in branched transport, and let $\beta=d\left(\alpha-\left(1-1/d\right)\right)\in(0,1)$, where $\alpha>1-1/d$ is the cost exponent. Write $\partial A$ for the boundary of $A$, and denote its Hausdorff and Minkowski dimensions by $\dim_H$ and $\dim_M$. Fractal dimension conjecture. The boundary satisfies

$$\dim_H(\partial A)=\dim_M(\partial A)=d-\beta.$$

The authors prove the upper bound on the Minkowski dimension, but not the matching lower bound. The conjecture predicts that the boundary has the exact, generally non-integer, dimension $d-\beta$.