Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2$.

Must $A *_C B$ have positive rank gradient?

The assumption on the index is necessary (otherwise we take $A,B = \mathbb{Z}/2\mathbb{Z}, C = \{1\}$).

I think that there should be a proof using Bass-Serre theory.