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There are cases in which this does hold. Greenberg proved that if $H_1$ and $H_2$ are Fuchsian groups with a common finite-index subgroup then each $H_i$ is of finite index in $\langle H_1,H_2\rangle$. I've no doubt that this is known more generally for quasiconvex subgroups of word-hyperbolic groups, although a reference currently eludes me.