Quantitative refinement of Pillai's conjecture

Let $\varepsilon>0$, and let $a,b,x,y$ be positive integers with $x,y\ge 2$ and $a^x\ne b^y$. Quantitative Pillai conjecture. There exists a constant $\kappa(\varepsilon)>0$ such that

$$|a^x-b^y|\ge \kappa(\varepsilon)\max\\{a^x,b^y\\}^{1-(1/x)-(1/y)-\varepsilon}.$$

This would give an effective lower bound for the distance between distinct perfect powers. It is presented as a conjectural quantitative strengthening of Pillai's conjecture and remains open.