Structural conjecture for maximal independent sets in generalized Kneser graphs

Let $d\ge 2$ be an integer, and let $q\Gamma_{2d+1,\{d,d+1\}}$ be the Kneser graph whose vertices are the relevant subspaces and whose adjacency is determined by dimensions $d$ and $d+1$. A point-pencil and a dual point-pencil are the standard maximal independent sets of these two types. Structural conjecture. For every integer $d\ge 2$ there is an integer $\rho(d)$ such that every maximal independent set of $q\Gamma_{2d+1,\{d,d+1\}}$ contains a point-pencil, a dual point-pencil, or has at most

$$\rho(d)\cdot q^{d^2+d-2}$$

elements.

This conjecture supplies the structural information on large maximal independent sets needed to extend the paper's chromatic-number argument from $d=3$ to every $d$ for which the corresponding result is available. Its resolution is not specified in the source.