The GCP(1) conjecture for missing digits and self-similar measures

Let $\mu$ be a missing digits measure on $\mathbb{R}$. Let $\dim_{\mathrm{H}}\mu$ denote its Hausdorff dimension, and let GCP(1) denote the good counting property with threshold $1$. A self-similar measure is assumed to satisfy the open set condition in the general statement.

GCP(1) conjecture. If

$$\dim_{\mathrm{H}}\mu>\frac{1}{2},$$

then $\mu$ has GCP(1). More generally, the same conclusion holds for self-similar measures satisfying the open set condition and having Hausdorff dimension greater than $1/2$.

The conjecture replaces an earlier $l^1$-dimension hypothesis by Hausdorff dimension and would characterize a broad class of measures with the good counting property. The source states it as a suspected conjecture and gives no resolution.