Cocliques in the Kneser graph on $(n-1,n)$-flags of PG$(2n,q)$

In the finite projective space PG$(2n,q)$ we consider flags of type $(n-1,n)$, that is, pairs $(A,B)$ consisting of an $(n-1)$-space $A$ and an $n$-space $B$ that are incident. Two such flags $(A_1,B_1)$ and $(A_2,B_2)$ are opposite if $A_1\cap B_2=A_2\cap B_1=\mathrm{\varnothing }$. Let ${\mathrm{\Gamma }}_{2n}$ be the graph whose vertices are the flags of type $(n-1,n)$ of PG$(2n,q)$, with two vertices being adjacent if the corresponding flags are opposite. Using the Erdős-Matching theorem for vector spaces shown by Ihringer, we determine, for $q$ large enough, the largest cocliques of ${\mathrm{\Gamma }}_{2n}$ and obtain a stability result. This EKR-type theorem proves a conjecture of D’haeseleer, Metsch and Werner.