The BEV bound for linear codes in projective spaces

Let $\mathbb{P}_q(n)$ be the projective space of subspaces of $\mathbb{F}_q^n$, equipped with the notion of linear code introduced by Braun, Etzion, and Vardy. The BEV bound states that the maximum size of a linear code in $\mathbb{P}_q(n)$ is

$$2^n.$$

Braun, Etzion, and Vardy conjectured this as the projective-space analogue of the $2^n$ bound for binary linear block codes. The paper notes that some linear codes not closed under intersection attain this bound, while whether it holds for all linear codes remains open.