A weak XYZ conjecture for ABC solutions

Let $\frak X$ be the set of primitive nonzero integer solutions $(A,B,C)$ to $A+B+C=0$, and let $H(A,B,C)=\max\{|A|,|B|,|C|\}$ and $S(A,B,C)=P^+(ABC)$.

Weak XYZ conjecture. For every $\delta>0$, there are only finitely many solutions $(A,B,C)\in\frak X$ for which

$$\log H(A,B,C)>\delta S(A,B,C).$$

The source presents this as a weak version of part (a) of the XYZ conjecture and explicitly says that, at the time of writing, it was not known to follow from the ABC conjecture.