Finiteness conjecture for restricted-digit additive triples

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

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

then there are only finitely many integers $(x,y,z)\in B_a\times B_b\times B_c$ such that

$$x+y=z.$$

The conjecture predicts finiteness in the dimension-sum-below-one regime. The paper presents it as a suspected consequence of the thinness of the restricted-digit sets; related results establish strong upper bounds, but the asserted finiteness is not supplied as a theorem here.