If a group $G$ acts on a CAT(0) cube complex, then does $G$ act on a simplicial tree?

There exist plenty of groups satisfying Serre's property (FA), meaning that any action on a simplicial tree has a global fixed point, which act nicely on a CAT(0) cube complex. It includes:

- Many Coxeter groups. Indeed, a group generated by torsion elements such that the product of any two of these elements has finite order satisfies Serre's property (FA). On the other hand, Coxeter groups act properly on finite-dimensional CAT(0) cube complexes.
- Thompson's groups T and V satisfy Serre's property (FA) but they act freely and properly on infinite-dimensional CAT(0) cube complexes.
- Random groups satisfy Serre's property (FA) but they acts geometrically on CAT(0) cube complexes.
- You can also construct other examples by using wreath products.

The most elementary example seems to be the triangle group $$T= \langle a,b,c \mid a^2=b^2=c^2=(ab)^3=(ac)^3=(bc)^3=1 \rangle.$$ It satisfies Serre's property (FA) since it is generated by torsion elements such that the product of any two of these elements has finite order. On the other hand, $T$ is the symmetry group of the tesselation of the plane by equilateral triangles. It naturally defines a collection of walls on the plane, so that by cubulating this wallspace, you deduce that $T$ acts properly on $\mathbb{R}^3$ (though of as a cube complex by looking at its usual tesselation with cubes).