Furstenberg-type restricted-digit counting conjecture

Let $p_1,\dots,p_k$ be $k\geq 2$ different odd prime numbers. For each $i\in\\{1,\dots,k\\}$, set

$$B_i=\\{0,\dots,(p_i-1)/2\\}.$$

Define

s=\sum_{i=1}^k\frac{\log\\#B_i}{\log p_i}=\sum_{i=1}^k\frac{\log(p_i+1)-\log 2}{\log p_i}.

Here $N_{p_1,\dots,p_k}^{B_1,\dots,B_k}$ denotes the set of nonnegative integers whose base-$p_i$ expansions use only digits from $B_i$ for every $i$. Restricted-digit counting conjecture. If $s\in(k-1,k)$, then for every $\epsilon>0$ there is a constant $C_\epsilon>1$ such that

C_\epsilon^{-1}N^{s-(k-1)-\epsilon}\leq\\#\left(N_{p_1,\dots,p_k}^{B_1,\dots,B_k}\cap[1,N]\right)\leq C_\epsilon N^{s-(k-1)+\epsilon}

for all integers $N\geq2$. If $s<k-1$, then $N_{p_1,\dots,p_k}^{B_1,\dots,B_k}$ is finite. The upper bound is known under Schanuel's conjecture, while the lower bound and finiteness assertion remain open; the claim is related to Furstenberg-type intersection problems.