Eventual unbounded growth of the generalized prime-factor function

Let $a_\alpha(n)$ be the function studied in the paper for a real parameter $\alpha>1$.

The generalized growth conjecture. For any $\alpha>1$ and $C>0$, for sufficiently large $n$ one has

$$a_{\alpha}(n+1)>a_{\alpha}(n)+C.$$

This would strengthen the expected eventual strict increase of $a_\alpha(n)$ by asserting arbitrarily large increments, but no resolution is given here.