Finiteness conjecture for intersections of restricted-digit sets

For an integer $b\geq 3$, let $B_b$ be the set of positive integers whose base-$b$ expansion contains only the digits $0$ and $1$. Let $a,b$ be multiplicatively independent integers, meaning that $\log a/\log b$ is irrational. Restricted-digit intersection conjecture. If

$$\frac{\log 2}{\log a}+\frac{\log 2}{\log b}<1,$$

then

$$\#(B_a\cap B_b)<\infty.$$

This is presented as a natural related problem and is attributed in the source to further discussion in BHY19. The finiteness of this intersection is not proved in the supplied text.