A possible formulation of the Bestvina-Feighn theorem is as follows (taken from here):
Combination Theorem (Bestvina & Feighn): If $H$ is a malnormal subgroup of hyperbolic groups $G_1, G_2$, then $G_1\ast_H G_2$ is hyperbolic.
I was wondering if the following variation is also true.
Question: Suppose that $G_1,G_2$ are hyperbolic groups with a common subgroup $H$ such that $H \leq G_1$ is malnormal while $H \leq G_2$ is finite index. Then is the amalgamated free product $G_1\ast_H G_2$ hyperbolic? What if we add in the hypothesis that $H$ is also quasi-convex in $G_1$ ?