On the exceptional set in the $abc$ conjecture

The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\mathrm{rad}(abc)<c^{1-\epsilon }$ for any $\epsilon >0$. Using the optimized methods in a recent work of Browning, Lichtman and Teräväinen, we showed that the number of those triples with $c⩽X$ is $O\left(X^{56\mathrm{/}85+\epsilon }\right)$ for any $\epsilon >0$, where $\frac{56}{85}\approx 0.658824$. This constitutes an improvement of the previous bound $O\left(X^{33\mathrm{/}50}\right)$.