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

Let $J_q(n,k)$ denote the Grassmann graph of $k$-dimensional subspaces of an $n$-dimensional vector space over the finite field with $q$ elements. A Boolean degree $1$ function is called trivial if it is one of the standard functions induced by a constant, a point, a hyperplane, or a point and a hyperplane in the sense described in the paper.

Eventual Grassmann graph classification conjecture. Let $q$ be a prime power. Then there exists a constant $n_q$ such that a Boolean degree $1$ function on $J_q(n,k)$ is trivial for all $n\geq n_q$ if $k,n-k\geq 2$.

This predicts that, for each fixed field size, all Boolean degree $1$ functions on Grassmann graphs are eventually among the standard examples as the ambient dimension grows. The paper presents this as a broader conjectural direction beyond the cases it proves.