Existence of length 2m−4, dimension 3 rank-metric intersecting codes for even m

Determine whether, for even integers m and any prime power q, there exists a nondegenerate [2m−4,3,d]_{q^m/q} rank-metric intersecting code for some minimum distance d.

Background

For m=6 the authors summarize known existence and non-existence results for [n,3]_{q^6/q} rank-metric intersecting codes: such codes exist for 5≤n≤7 and n=9, and do not exist for n<5 or n>9. The n=8 case remained unresolved in Section 5.

In the Conclusions, the authors elevate this to a broader open problem, stating that the existence of [2m−4,3]_{q^m/q} rank-metric intersecting codes is open even when m is even.