Sharp exponent conjecture for restricted-digit additive triples

For an integer $b\geq 3$, let $B_b$ be the set of nonnegative integers whose base-$b$ expansion contains only the digits $0$ and $1$. Let $a,b,c\geq 3$ be pairwise multiplicatively independent integers, and set

$$s=\frac{\log 2}{\log a}+\frac{\log 2}{\log b}+\frac{\log 2}{\log c}.$$

Sharp-exponent conjecture. If $s>1$, then for each $\epsilon>0$ there exists a constant $C$ such that, for every integer $N$, the number of solutions $(x,y,z)$ with $x,y,z\geq 0$, $z\leq N$, and $x\in B_a$, $y\in B_b$, $z\in B_c$ is at most $CN^{s-1+\epsilon}$ and at least $C^{-1}N^{s-1-\epsilon}$.

This asks whether the exponent $s-1$ in the paper's upper bound is essentially sharp. The source describes the corresponding upper estimate as obtainable by a dimension-decomposition method, while the matching lower bound remains conjectural.