Exact maximization by the cyclic regular tournament

Establish whether, for every sufficiently large odd integer n, the cyclic regular tournament T_n with vertex set Z/nZ and edges i→j whenever j−i mod n is less than n/2 attains the exact maximum number C(n) of directed Hamiltonian cycles among all tournaments on n vertices.

Background

The paper proves that the cyclic regular tournament T_n is an asymptotic maximizer: its number of Hamiltonian cycles is asymptotic to the maximum possible value C(n). However, this asymptotic result does not establish exact optimality for any sufficiently large finite odd order.

The authors note that exact optimality fails for small orders, giving the example that the Paley tournament Q_7 has 24 Hamiltonian cycles whereas T_7 has only 17. The unresolved conjecture asks whether this discrepancy disappears for all sufficiently large odd n.