The missing-digits product-set interval conjecture

Let $K$ be a missing digits set, meaning a compact set determined by restricting the digits in a fixed integer base. Folklore conjecture. If

$$\dim_{\mathrm{H}} K>1/2,$$

then $K\cdot K$ contains intervals. This is presented as a folklore conjecture, and Theorem 3 can be viewed as a partial answer; its general status is not resolved in the supplied text.