Wu's divisibility-by-3 conjecture for symmetric concatenations

Let $a(n)$ be the smallest prime obtained by inserting $n$ between two copies of a positive-length string of identical decimal digits, or $0$ if no such prime exists. A positive integer is a multiple of $11$ when it is divisible by $11$.

Wu's divisibility-by-3 conjecture. If $n$ is not a multiple of $11$ with an even number of digits and $a(n)=0$, then $n$ is a multiple of $3$.

This is proposed after the original OEIS conjecture has been shown false. The paper reports computational examples supporting the claim, but the supplied text gives no resolution.