Define $\text{rad}_{23}(2^m3^nr)=2^{\text{sign}(m)}3^{\text{sign}(n)}r$, where $m,n\ge0$ and $2,3\nmid r\in\mathbb{N}$.

For a triple $a+b=c$ define the quality $q_{23}(a,b,c)=\frac{\log(c)}{\log(\text{rad}_{23}(abc))}$.

Has anyone attempted to specifically prove that only finitely many primitive triples have quality $q_{23}>1+\epsilon$ for any given $\epsilon>0$? Is there reason to hope this may be within reach of well established methods?

Based on this table of $abc$-triples, only 2 mini-$abc$-triples: are known with quality $\ge1.4$:

$37 + 2^{15} = 3^8 \times 5,\ \ \ \ \ \ \ q_{23}=1.48291$

$5 + 3^{11} = 2^{10} \times 173,\ \ \ \ \ q_{23}=1.41268$