The base-subset collineation conjecture for polar Grassmannians

Let ${\mathcal G}_k$ be the Grassmannian of $k$-dimensional singular subspaces of a building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$, and let ${\mathfrak G}_k$ be its shadow space. A base-subset collineation conjecture asserts that every bijective transformation of ${\mathcal G}_k$ sending the shadows of apartments to the shadows of apartments is a collineation of ${\mathfrak G}_k$. The statement concerns the relation between apartment shadows and the incidence geometry of polar Grassmannians; the supplied text does not indicate whether it has been proved or disproved.