The large-prime 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 large-prime valuation conjecture. If

$$p\in[\sqrt{k},k)$$

is a prime, then

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

This is a refinement of the preceding conjecture for primes in the interval $[\sqrt{k},k)$, proposed on the basis of numerical computations. A closed formula for the repair factor is not expected to be readily accessible.