The eventual classification conjecture for Boolean degree 1 functions on polar graphs

Let $C_q(n,k,e)$ denote the polar graph considered in the paper, whose vertices are the relevant $k$-dimensional subspaces of a polar space over the finite field with $q$ elements. A Boolean degree $1$ function is called trivial when it is induced from the standard functions on the ambient Grassmann graph: constants, point functions, hyperplane functions, and their combinations.

Polar graph classification conjecture. Let $k\geq 2$. Then there exists $n_{q,k}$ such that every Boolean degree $1$ function $f$ on $C_q(n,k,e)$ with $n\geq n_{q,k}$ is trivial, that is, $f$ can be written as

$$f^\pm=\bigvee_i p_i^+\vee\bigvee_i\pi_i^+\vee\bigvee_i(\tilde{\pi}_i^+\wedge\tilde{p}_i^-).$$

Here $p_i$ are points of the polar space, $\pi_i$ are non-degenerate hyperplanes of the ambient space, and $\tilde{\pi}_i$ are degenerate hyperplanes of the ambient space with $\tilde{\pi}_i=\tilde{p}_i^\perp$. The conjecture seeks an eventual classification of Boolean degree $1$ functions on polar graphs; the paper states that even their general description appears complicated and proves only a particular case.