Approximate Hamilton packing conjecture for regular tripartite digraphs

Let $G$ be a $cn$-regular tripartite digraph with three vertex classes, each of size $n$, meaning that every vertex has indegree and outdegree $cn$. A Hamilton cycle is a directed cycle containing all $3n$ vertices. Tripartite approximate packing conjecture. Let $\varepsilon>0$, $c>1$, and let $n$ be sufficiently large. Then $G$ contains at least $(1-\varepsilon)cn$ edge-disjoint Hamilton cycles. This is proposed as an approximate version of the conjecture that every regular tripartite tournament has a Hamilton cycle decomposition; neither the exact decomposition conjecture nor this approximate version is proved here.