Conjecture A on the refined abc bound

For coprime positive integers $a$ and $b$, put $c=a+b$ and define

$$\gamma=\gamma(abc),$$

where $\gamma(n)=\prod_{p\mid n}p$ is the radical of $n$. Conjecture A. There exists a real number $C_1$ such that

$$c<\gamma\exp\left(4\sqrt{\frac{3\log\gamma}{\log\log\gamma}}\left(1+\frac{\log\log\log\gamma}{2\log\log\gamma}+\frac{C_1}{\log\log\gamma}\right)\right).$$

Furthermore, there exists a real number $C_2$ and infinitely many pairs of coprime positive integers $a$ and $b$ for which

$$c>\gamma\exp\left(4\sqrt{\frac{3\log\gamma}{\log\log\gamma}}\left(1+\frac{\log\log\log\gamma}{2\log\log\gamma}+\frac{C_2}{\log\log\gamma}\right)\right).$$

This is presented as the most precise refinement of the function $C(\epsilon)\gamma(abc)^{1+\epsilon}$ known to the source, and is attributed there to Conjecture A of Ochoa, de la Bretèche and Granville. Its resolution status is not specified in the supplied text.