Packing edge-disjoint triangles in regular and almost regular tournaments

For a tournament T, let ^ν3(T) denote the maximum number of pairwise edge-disjoint triangles (directed cycles of length 3) in T. Let ^ν3(n) denote the minimum of ^ν3(T) ranging over all regular tournaments with n vertices (n odd). We conjecture that ^ν3(n)=(1+o(1))ⁿ²/9 and prove thatⁿ²11.43(1-o(1))≤^ν3(n)≤ⁿ²9(1+o(1)) improving upon the best known upper bound of ⁿ²-18 and lower bound of ⁿ²11.5(1-o(1)). The result is generalized to tournaments where the indegree and outdegree at each vertex may differ by at most βn.