The Hermitian forms graph classification conjecture

Let $H_q(\ell,k)$ denote the Hermitian forms graph, identified with the $k$-dimensional subspaces of the Grassmann graph $J_q(k+\ell,k)$ that are disjoint from a fixed $\ell$-space $L$. A Boolean degree $1$ function is called trivial when it is induced by point and hyperplane functions of the ambient Grassmann graph.

Hermitian forms graph classification conjecture. Let $q$ be a prime power. Let $f$ be a Boolean degree $1$ function on $H_q(\ell,k)$, where $k+\ell$ is sufficiently large depending on $q$. Then there exist a line $g$ meeting $L$ in a point, an $(\ell-1)$-space $G\subseteq L$, points $p_i\in g$, and hyperplanes $\pi_i$ with $\pi_i\cap L=G$, and with $p_j\notin\pi_i$ for all $p_i$ and $\pi_j$, such that

$$f^\pm=\bigvee p_i^+\vee\bigvee\pi_i^+.$$

The conjecture proposes a complete description of Boolean degree $1$ functions on Hermitian forms graphs in sufficiently large dimension. The paper gives the listed induced examples and leaves their classification for future work.