The extremal arc-count conjecture for strongly connected digraphs without acyclic separators

Let $D$ be a strongly connected digraph with $n\geq 3$ vertices and $m$ arcs. An acyclic separator is a separator whose induced subdigraph is acyclic. Extremal arc-count conjecture. If $D$ does not have an acyclic separator, then

$$m\geq 3n-3.$$

The digraph $\vec{C}_n^*$ described in the surrounding text has $3(n-1)=3n-3$ arcs and is strongly $2$-connected, neighborhood-cyclic, and has no acyclic separator, motivating the claimed sharp lower bound.