The converse to uniform complexity implying dimension equality for sofic self-affine fractals

Let $S \subset D_2^{\mathbb{N}}$ be a sofic system with primitive adjacency matrix. Let $X$ be the associated self-affine fractal, and say that $S$ has uniform complexity when the uniform complexity condition defined in the paper holds. Uniform-complexity converse conjecture. The Hausdorff and Minkowski dimensions of $X$ are equal if and only if $S$ has uniform complexity:

$$\mathrm{dim}_{\mathrm{H}}(X)=\mathrm{dim}_{\mathrm{M}}(X) \quad\Longleftrightarrow\quad S\text{ has uniform complexity}.$$

The forward implication is the conjectured converse to the preceding corollary, while the reverse implication is established there. The claim concerns when these two dimension notions coincide for sofic self-affine fractals with primitive adjacency matrix.