On the largest independent sets in the Kneser graph on chambers of PG(4,q)

Let Γ₄ be the graph whose vertices are the chambers of the finite projective $4$-space PG(4,q), with two vertices being adjacent if the corresponding chambers are in general position. For q≥ 749 we show that α:=(q²+q+1)(q³+2q²+q+1)(q+1)² is the independence number of Γ₄ and the geometric structure of independent sets with α vertices is described.