Non-rigidity conjecture for real projective spaces

Let $n\geqslant 4$ and let $G\subset\mathrm{PGL}_{n+1}(\mathbb{R})$ be a finite subgroup. The projective space $\mathbb{P}^n_{\mathbb{R}}$ is called $G$-birationally rigid when it is birationally rigid with respect to the action of $G$.

Non-rigidity conjecture. For every $n\geqslant 4$ and every finite subgroup $G\subset\mathrm{PGL}_{n+1}(\mathbb{R})$, $\mathbb{P}^n_{\mathbb{R}}$ is not $G$-birationally rigid.

The conjecture extends the theorem established in the paper for $n\in\{4,5,6\}$ and predicts non-rigidity for all higher-dimensional real projective spaces. The context states that no examples of finite groups yielding $G$-birational rigidity are known in the corresponding complex dimensions $n\geqslant 5$, so the general assertion remains open.