The classification of digraphs of second minimal odd-period orbits

A second minimal $(2k+1)$-orbit is an orbit of odd period $2k+1$ occupying the second level in the relevant minimal-orbit ordering, and its digraph is the directed graph associated with that orbit. Two digraph types are identified when they are inverses of one another.

Second minimal odd-orbit digraph conjecture. For every integer $k\geq 3$, the digraph of any second minimal $(2k+1)$-orbit has one of $4k-3$ possible types up to inverses.

The method proving the classification for second minimal $7$-orbits extends to show that second minimal $9$-, $11$-, and $13$-orbits have respectively $13$, $17$, and $21$ possible types up to inverses. The conjecture predicts the general count for all odd periods at least $7$.