The first non-trivial primitive number for repunit parameters

Let $m$ be the number whose base-$g$ expansion consists of $g$ digits equal to $1$, and let $g=p+1$ where $p$ is a prime number. Primitive-number conjecture. Then $m$ is the first non-trivial primitive number. The statement concerns the ordering of primitive numbers associated with the base-$g$ digit system; the supplied text gives no evidence about whether it has been proved or disproved.