A bounded-limit conjecture for the generalized quality metric

For an $abc$-triple $(a,b,c)$, let $q_C(a,b,c;\alpha,\beta)$ be the alternative quality metric considered in the source, with parameters $\alpha$ and $\beta$. Generalized quality metric conjecture. For every $\alpha\geq0$, there exists $\beta=\beta(\alpha)>0$ such that

$$\limsup_{c\to\infty}\max\{q_C(a,b,c;\alpha,\beta):(a,b,c)\text{ is an }abc\text{-triple}\}=L$$

for some finite, nonzero $L$. This conjecture proposes that each fixed value of $\alpha$ admits a choice of $\beta$ producing a finite positive asymptotic upper envelope; the supplied source gives no resolution.