HAW conjecture for bounded diagonal rays on homogeneous spaces

Let $G$ be a Lie group, let $\Gamma\subseteq G$ be a lattice, and set $X=G/\Gamma$. Let $\mathbf{P}^+(\mathfrak{g})=(\mathfrak{g}\smallsetminus\{0\})/\sim$, where $\mathbf{v}\sim\mathbf{v}'$ if and only if $\mathbf{v}\in\mathbb{R}^+\mathbf{v}'$, and for $x\in X$ define

$$E^+(x,\infty):=\{[\mathbf{v}]\in\mathbf{P}^+(\mathfrak{g}):\{\exp(t\mathbf{v})x:t\geq 0\}\text{ is bounded in }X\}.$$

HAW conjecture. For any $x\in X$, the set $E^+(x,\infty)$ is hyperplane absolute winning (HAW) on $\mathbf{P}^+(\mathfrak{g})$. This is the homogeneous-space counterpart of the preceding conjectural strengthening of thickness for bounded geodesic directions; the supplied source does not state a resolution.