Euler-bound analogue for fixed k≥3

Determine whether, for each fixed integer k≥3 and all sufficiently large n, the orientable genus of the bipartite Kneser graph H(n,k) equals the rounded Euler lower bound.

Background

For k≥3, the paper notes that substantially less is known. In particular, the available computations give 85≤γ(H(8,3))≤143 and 337≤γ(H(9,3))≤338, with the lower endpoint in each case equal to the Euler bound.

The authors identify the corresponding large-n statement for each fixed k≥3 as a natural unresolved analogue of the conjecture for H(n,2). They also note that the symmetry used in the paper is generally unavailable: Godsil's classification permits a subgroup acting simply transitively on the k-subsets only for (n,k)=(8,3) or (32,3), indicating that new constructions are needed.