Lagarias–Soundararajan's strong 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 define

$$H(A,B,C)=\max\{|A|,|B|,|C|\},\qquad S(A,B,C)=P^+(ABC),$$

where $P^+(n)$ is the largest prime divisor of $n$.

XYZ conjecture (strong form). For every $\epsilon>0$, both of the following hold:

(a) only finitely many $(A,B,C)\in\frak X$ satisfy

$$\log H(A,B,C)>S(A,B,C)^{2/3+\epsilon};$$

(b) infinitely many $(A,B,C)\in\frak X$ satisfy

$$\log H(A,B,C)>S(A,B,C)^{2/3-\epsilon}.$$

This conjecture relates the height of an ABC solution to the largest prime factor of its product. The text says that the ABC conjecture implies a weaker version of part (a), with exponent $1+\epsilon$, while the stated exponent $2/3+\epsilon$ is not known to follow from ABC.