The prime-power repair-factor conjecture for Stirling sequences

Let $S^{(2)}_k$ denote the relevant sequence and let ${\rm Fail}(S^{(2)}_k)$ be its repair factor. For a prime $p$ and a positive integer $j$, the following assertions are conjectured. The prime-power repair-factor conjecture.

• If $j=1$, then

$${\rm Fail}\bigl(S^{(2)}_{p+1}\bigr)=p\cdot{\rm Fail}\bigl(S^{(2)}_p\bigr).$$

• If $j>1$, then

$${\rm Fail}\bigl(S^{(2)}_{p^j+1}\bigr)\mid p^{j-1}\cdot{\rm Fail}\bigl(S^{(2)}_{p^j}\bigr).$$

• If $j>1$, then

$$\bigl|{\rm Fail}\bigl(S^{(2)}_{p^j}\bigr)\bigr|_p=p^{-1}.$$

• We have

$$\bigl|{\rm Fail}\bigl(S^{(2)}_{p^j+1}\bigr)\bigr|_p=p^{-j}.$$

These are the prime-power cases of the conjectural behavior of the repair factor and its $p$-adic valuation. They, like the preceding conjectures, are motivated by numerical computations, while a general closed formula for the repair factor appears inaccessible.