Continuity of the renormalised volume between Epstein surfaces with respect to the Weil--Petersson topology

We prove that the renormalised volume between the two Epstein surfaces generated by a Weil–Petersson quasicircle is continuous with respect to the Weil–Petersson topology. As a consequence, we prove that the universal Liouville action is proportional to this renormalised volume for any Weil–Petersson quasicircle.