HAW conjecture on bounded points in the expanding subgroup

Let $G$, $\Gamma$, $X$, and $F^+$ be as in the HAW conjecture for bounded orbits of $\operatorname{Ad}$-diagonalizable flows. Let $H=H(F^+)$ be the subgroup defined in the source. Consider the set

$$\{h\in H:h\Gamma\in E(F^+)\}.$$

HAW conjecture on $H$. The set

$$\{h\in H:h\Gamma\in E(F^+)\}$$

is HAW on $H$. This is proposed as a conjectural generalization of the corresponding HAW theorem and would imply the preceding conjecture. The analogous property is known in the specific case discussed in the source, but the stated generalization remains open.