The radical-growth conjecture for the repair factor of Stirling sequences

Let $S^{(2)}_k$ denote the relevant sequence, let ${\rm Fail}(S^{(2)}_k)$ be its repair factor, and let ${\rm Rad}(n)$ denote the radical of $n$, the product of its distinct prime divisors. The radical-growth conjecture. The repair factor asymptotically involves non-trivial powers of prime divisors of $(k-1)!$, in the sense that

$$\lim_{k\to\infty}\frac{{\rm Fail}(S^{(2)}_k)}{{\rm Rad}\bigl((k-1)!\bigr)}=\infty.$$

This is one of four conjectures arising from numerical computations in a setting where a closed formula for the repair factor seems unlikely to be accessible. It predicts that the repair factor eventually grows beyond the radical of $(k-1)!$.