Homogeneous-fibre conjecture for stationary measures in Case 2.3

Let $G=\operatorname{PGL}_n(\mathbb{R})$. Let $X$ and the groups $Q$, $R_0$, $R$, $S\simeq Q/R_0$, $\Lambda\simeq R/R_0$, and $H$ be as defined above. Suppose that we are in Case 2.3: $Q_H$ is a parabolic subgroup of $H$ and

$$Q_H^o\cap R_0=\{\operatorname{id}\}.$$

Let $\nu$ be a $\mu$-stationary and ergodic probability measure on $X_\mathcal{C}$. Homogeneous-fibre conjecture. There exists a trivialization $X_\mathcal{C}\simeq H/Q_H\times S/\Lambda$ in whose coordinates $\nu$ is a product measure $\overline{\nu}_F\otimes\tilde{\nu}$, where $\tilde{\nu}$ is $S_0$-homogeneous. Equivalently, the fibre measures are homogeneous, without assuming that the $H$-action on $\mathbb{P}(\mathbb{R}^n)$ is irreducible. This extends the homogeneous-fibre conclusion known under the irreducibility hypothesis; the conjecture concerns the remaining Case 2.3.b.