Approximate Hamilton packing conjecture for diregular bipartite tournaments

Let $G$ be a diregular bipartite tournament on $2n$ vertices, so $G$ is a complete bipartite orientation with equal indegree and outdegree at every vertex. Edge-disjoint Hamilton cycles are Hamilton cycles having no common edge. Approximate packing conjecture. Let $\varepsilon>0$ and let $n$ be sufficiently large. Then every diregular bipartite tournament on $2n$ vertices contains at least $(1/2-\varepsilon)n$ edge-disjoint Hamilton cycles. This would give an asymptotically optimal packing, since a diregular bipartite tournament has $n/2$ outgoing edges at each vertex; the paper does not establish this bound.