Braun–Etzion–Vardy conjecture on the maximum size of binary linear projective-space codes

Let $\mathbb{P}_2(n)$ be the projective space consisting of the subspaces of $\mathbb{F}_2^n$, and let a linear code be a subset of $\mathbb{P}_2(n)$ endowed with the linear structure considered for codes in projective spaces. Braun–Etzion–Vardy conjecture. The maximum size of a linear code in $\mathbb{P}_2(n)$ is

$$2^n.$$

This conjecture concerns the largest projective-space analogue of a binary linear block code. The supplied context does not establish whether the conjecture has been resolved.