Constant-depth decision-tree conjecture for Boolean functions on Grassmann graphs

Let $J_q(n,k)$ be the Grassmann graph on the $k$-dimensional subspaces of an $n$-dimensional vector space over the finite field of order $q$. A Boolean degree $d$ function is a Boolean function of degree $d$ in the Grassmann association scheme. Constant-depth decision-tree conjecture. For every given $q$ and $d$, there exists a $k_0$ such that if $k,n-k\geq k_0$, then every Boolean degree $d$ function on $J_q(n,k)$ is a constant-depth decision tree, with queries asking whether a point is contained in a subspace or whether a subspace is contained in a hyperplane. The paper presents this as a conjecture motivated by examples; no resolution is supplied.