Finiteness of Cannon--Thurston fibers

Let $Y\to X$ be a proper map between proper hyperbolic metric spaces. A Cannon–Thurston map is a continuous extension $\mathrm{\partial }Y\to \mathrm{\partial }X$. We prove that in most known settings in which a Cannon–Thurston map exists it is uniformly finite-to-one. This answers a question due to Swarup from Bestvina’s problem list and generalizes previous results of Cannon–Thurston, Kapovich–Lustig, Dowdall–Kapovich–Taylor and Ghosh.