The $p$-adic valuation conjecture for the repair factor of Stirling sequences

Let $S^{(2)}_k$ denote the relevant sequence and let ${\rm Fail}(S^{(2)}_k)$ be its repair factor. For a prime $p$, write $|\cdot|_p$ for the $p$-adic absolute value. The $p$-adic valuation conjecture. If $p<k$ is a prime, then

$$|{\rm Fail}(S^{(2)}_k)|_p<1.$$

This is the first of four conjectures suggested by numerical computations because a closed formula for the repair factor seems unlikely to be accessible. The conjecture concerns the divisibility of the repair factor by primes smaller than the index.