Finiteness conjecture for the closed geodesic norm on higher-rank locally symmetric spaces

Let $M=\Gamma \backslash G/K$ be a locally symmetric space, where $G$ is a semisimple Lie group, $K\subset G$ is a maximal compact subgroup, and $\Gamma\subset G$ is a lattice. Assume that $M$ has rank at least $2$ and that $\Gamma$ is invariant under the Cartan involution. The closed geodesic norm is the word norm on the subgroup generated by elements represented by closed local geodesics, with respect to the generating set of elements represented by closed geodesics.

Finiteness conjecture. The diameter of the closed geodesic norm is finite.

The conjecture seeks a direct geometric generalization of the known finiteness result for manifolds whose fundamental group is virtually an irreducible $S$-arithmetic Chevalley group of rank at least $2$. Equivalently, the associated word norm on $\Gamma$ generated by elements invariant under the Cartan involution should have finite diameter.