The exact-overlaps conjecture for self-similar sets

Let $X$ be the attractor of a finite iterated function system of contracting similarities of $\mathbb{R}]$ having common contraction ratio $r$, with defining map set indexed by a finite set $\Lambda$. Write $\operatorname{\dim_\textrm{B}}(X)$ for the box-counting dimension. The standard covering argument gives

$$\operatorname{\dim_\textrm{B}}(X)\leq\min\left\{1,\frac{\log|\Lambda|}{\log(1/r)}\right\}.$$

An exact overlap occurs when distinct words $\underline{i},\underline{j}\in\Lambda^n$ define the same iterated map, $f_{\underline{i}}=f_{\underline{j}}$.

Exact-overlaps conjecture. A strict inequality in the preceding box-dimension bound can occur only in the presence of exact overlaps.

The conjecture asserts that persistent dimension loss cannot arise merely from approximate alignment of cylinder sets. It appears in a question of Peres and Solomyak and includes important special cases such as Furstenberg's projection problem for the one-dimensional Sierpiński gasket and the $0,1,3$ problem; the corresponding question for self-similar measures includes the Bernoulli convolutions problem. The source gives no resolution, so the conjecture remains open.