Strong Property (T) elementary action conjecture for Gromov-hyperbolic spaces

Let $G$ be a discrete countable group with strong Property $(T)$, and let $(X,d)$ be a Gromov-hyperbolic space.

Strong Property (T) action conjecture. Any action of $G$ by isometries on $X$ is elementary.

This would extend the known bounded-orbit result for actions on uniformly locally finite Gromov-hyperbolic graphs. It is open, including the question of whether mapping class groups of closed oriented surfaces have strong Property $(T)$.