Converse-invariant orientations of trees with maximum degree at least three

Let $D$ be an orientation of a tree with maximum degree at least $3$. Write $-D$ for the converse orientation, and say that $D$ is converse invariant when it has the same number of copies in every tournament as $-D$. The bridge-mirroring operation is the operation defined in the paper that mirrors an oriented bridge; recursively applying it produces orientations from an orientation of a path.

Conjecture on converse-invariant tree orientations. $D$ is converse invariant if and only if

$$D\cong -D$$

or $D$ can be obtained by applying the bridge-mirroring operation recursively to an orientation of a path.

The paper establishes this characterization for orientations of trees with diameter bounded by three. The conjecture proposes that the same description holds for all orientations of trees with maximum degree at least $3$.