Infinitely many non-power lengths with universal invertibility of aperiodic necklaces

Let $p$ be a positive prime number. An aperiodic necklace of length $n$ over $\Sigma_p$ is understood in the sense used in the paper, and its weight is taken modulo $p$.

Necklace invertibility conjecture. There exist infinitely many $n$, different from a power of $p$, for which every aperiodic necklace of length $n$ over $\Sigma_p$ with non-zero weight modulo $p$ is invertible.

By the preceding corollary, this is equivalent to the assertion that for each prime $p$ there are infinitely many $p$-rooted primes. It is motivated by Artin's conjecture on primitive roots and concerns lengths for which all relevant necklaces, or equivalently the corresponding generalized de Bruijn words, have the required invertibility property.