Hansen–Johnsen–Ranestad conjecture on higher weights of Grassmann codes

Let $G(l,m)$ be the Grassmann variety of $l$-dimensional subspaces of an $m$-dimensional vector space over ${\mathbb F}_q$, embedded in the Plücker projective space $\mathbf P^{k-1}$, where $k=\binom{m}{l}$. Let $d_i$ be the higher weights of the Grassmann code $C(l,m)$, and let $J_i$ be the maximal number of ${\mathbb F}_q$-rational points of $G(l,m)$ lying on a codimension $i$ linear subspace of the Plücker space. A Schubert union is a union of Schubert varieties in $G(l,m)$. Hansen–Johnsen–Ranestad's conjecture. The higher weights of the Grassmann codes $G(l,m)$ are always computed by Schubert unions: for every $i$, there is a codimension $i$ linear subspace whose intersection with $G(l,m)$ is a Schubert union containing $J_i$ ${\mathbb F}_q$-rational points, and hence $d_i=n-J_i$. The conjecture predicts that Schubert unions suffice to attain the maxima governing all higher weights; the paper's stated goal is to confirm it in the case $l=2$, while the general assertion is presented as the conjecture from the cited work.