Let $A,B$ be finitely generated groups with a common **finite** subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$.

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

See Which 3-manifolds have positive rank gradient? for a definiton of rank gradient.

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

The case $C = \{1\}$ has been established by Marc Lackenby - Proposition 3.2 of http://people.maths.ox.ac.uk/lackenby/er190804.pdf

I think that there should be a proof using Bass-Serre theory, at least in the case that $A,B$ are residually finite. The argument is as follows:

Since $A,B$ are residually finite, $G = A *_C B$ is residually finite as well. It is enough to show that the rank of normal subgroups $N$ of finite index in $G$ grows linearly with the index $[G : N]$. Residual finiteness, allows us to take $N$ to be disjoint from $C$ in $G$. From Bass-Serre theory, one can hopefully write $G$ as a free product with amalgamation (since $N$ does not meet any conjugate of $C$ in $G$) and calculate its rank.