I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that
a) the graph $\Gamma$ is fine,
b) $\Gamma$ is not a tree,
c) not all edges have finite stabilizer,
d) the action of $G$ on $\Gamma$ is minimal, i.e. there is no proper, connected $G$-invariant subgraph of $\Gamma$.

Here I use the definition that $G$ acts acylindrical on $\Gamma$ iff there is a $k\in\mathbb{N}$ such that all geodesic segments of length $k$ in $\Gamma$ have finite (pointwise) stabilizer.

Examples are known if one condition is dropped: If I drop c), then any relative hyperbolic group (in the sense of Bowditch) is an example. If I drop b) it is easy enough to construct examples by taking say a malnormal subgroup $C\leq A$ and letting $A\ast_C B$ act on its Bass-Serre tree. If I drop a), then any acylindrically hyperbolic group will do by definition (at least if I drop cocompactly as well).

Does anyone know where to find an example in the literature satisfying a)-c) and/or how to construct one?

Edit (Oct 23rd 2015): Condition d) was added.
Edit (Oct 24th 2015): added "connected" in Condition d)