For a regular tournament T of odd order n, let cm(T) be the number of cycles of length m in T. It is well known according to U. Colombo (1964) that c4(T)≤c4(RLTn), where RLTn is the unique regular locally transitive tournament of order n. In turn, in 1968, A. Kotzig proved that c4(DRn)≤c4(T), where DRn is a doubly-regular tournament of order n. However, the spectral tools allow us to simply show that the converse inequality c5(RLTn)≤c5(T)≤c5(DRn) holds. Recently we have proved that c6(T)≤c6(DRn) and conjectured that c6(RLTn)≤c6(T). For these values of m, the same results can be also formulated for the trace trm(T) of the mth power of the adjacency matrix of T. (We consider this quantity here because it equals the number of closed walks of length m in T.) In the present paper, we determine c7(DRn) and c7(RLTn). Comparing c7(DRn) with c7(RLTn) yields the inequality c7(RLTn)<c7(DRn), while tr7(DRn)<tr7(RLTn) for n≥7. We also present some additional arguments which make it possible to suggest that for each odd n≥9, the two-sided bounds c7(RLTn)≤c7(T)≤c7(DRn) and tr7(DRn)≤tr7(T)≤tr7(RLTn) hold.