The linear-backwards-edge conjecture for forbidden-cycle oriented graphs and digraphs

Let $C_k$ be a directed cycle on $k$ vertices, with $k,n\in\mathbb{N}$ and $k\geq 3$. Given a labelled digraph or oriented graph $G$ on vertices $1,\dots,n$, a transitive-optimal ordering is an ordering of its vertices that minimises the number of backwards edges, where a backwards edge is directed from a vertex appearing later to one appearing earlier in the ordering.

Linear-backwards-edge conjecture. The following hold:

1. Almost all $C_k$-free oriented graphs on $n$ vertices have $\Theta(n)$ backwards edges in a transitive-optimal ordering.
2. If $k$ is even, then almost all $C_k$-free digraphs on $n$ vertices have $\Theta(n)$ backwards edges in a transitive-optimal ordering.

The preceding theorem establishes only weaker bounds, namely between $c n/\log n$ and $\alpha n^2$ in the oriented-graph case and, for even $k$, the same type of bounds in the digraph case. The conjecture strengthens these to linear order and remains open in the source.