HAW conjecture for bounded geodesics on locally symmetric spaces

Let $M$ be a locally symmetric space of non-compact type with finite Riemannian volume. For $p\in M$, let $S_p(M)$ denote the unit tangent sphere at $p$, and let $E^+(p,\infty)$ be the set of unit tangent directions whose forward geodesic rays remain bounded. HAW conjecture. For any $p\in M$, the set $E^+(p,\infty)$ is hyperplane absolute winning (HAW) on $S_p(M)$. This strengthens the preceding thickness result and would provide the stronger intersection and largeness properties associated with HAW sets; its status is not resolved in the supplied source.