Generalized Kollár–Peskine conjecture for infinite Grassmannians

For any $n\geq 2$, let $K$ and $R$ be the fields and rings used to define the infinite Grassmannian, and let $\mathcal{X}_n$ be the infinite Grassmannian associated to $G=\operatorname{SL}(n)$:

$$\mathcal{X}_n:=\operatorname{SL}(n,K)/\operatorname{SL}(n,R).$$

Generalized Kollár–Peskine conjecture. There does not exist any nonconstant morphism

$$\phi:\mathbb{P}^{n+1}\to\mathcal{X}_n.$$

This generalizes the preceding case for the infinite Grassmannian associated to $\operatorname{SL}(2)$ and would imply corresponding triviality results for vector bundles over projective space. The source provides no resolution of the conjecture.