Wu's divisibility-by-3 conjecture for appended-digit primes

Let $c(n)$ be the smallest prime obtained by appending a positive-length string of identical decimal digits to $n$, or $0$ if no such prime exists.

Wu's divisibility-by-3 conjecture. If $c(n)=0$, then $n$ is divisible by $3$.

The claim concerns the analogous one-sided digit-concatenation sequence. The supplied text gives no proof or resolution.