Uniqueness of the base-3 characterization of repunit primes in Cantor sets

For $N\in\mathbb{N}\setminus\{1\}$, call a prime $p$ a base-$N$ repunit prime if

$$(N-1)p+1=N^q$$

for a prime $q$. Let $\mathcal{C}_N$ denote the corresponding Cantor set obtained by retaining base-$N$ expansions using only the digits $0$ and $N-1$.

Uniqueness conjecture. It is possible to characterize base-$N$ repunit primes as precisely the primes whose reciprocals belong to $\mathcal{C}_N$ only when $N=3$; for every $N\neq 3$, this characterization does not hold.

The paper proves the characterization for $N=3$ and shows directly that it fails for $N=2$, while the conjecture concerns all other bases.