Algebraic diffeomorphism transitivity conjecture for rational surfaces

Let $X$ be a smooth projective rational surface, and let $n$ be a natural integer. The group of algebraic diffeomorphisms $\operatorname{Diff}_{\mathrm{alg}}(X(\mathbb{R}))$ consists of algebraic diffeomorphisms of the real locus $X(\mathbb{R})$. Algebraic diffeomorphism transitivity conjecture. The group $\operatorname{Diff}_{\mathrm{alg}}(X(\mathbb{R}))$ acts $n$-transitively on $X(\mathbb{R})$. The only evidence given is the case $X=\mathbb{P}^1\times\mathbb{P}^1$, established by the preceding theorem; the conjecture is intended to provide the transitivity needed to compare rational models of nonorientable surfaces, and its general case remains open.