Geodesic containment conjecture for the Tamari lattice

Let $T_1$ and $T_2$ be trees with an upper bound in the Tamari lattice $\mathscr{C}_n$, and let $\mathscr{R}_n$ be the rotation graph on these trees. A shortest path in $\mathscr{R}_n$ from $T_1$ to $T_2$ also lies in $\mathscr{C}_n$. This asserts that whenever the endpoints have a common upper bound in the Tamari lattice, an ambient shortest rotation path remains entirely within the Tamari lattice. The preceding result establishes the minimum path length and characterizes the moves in any shortest path, but the supplied text does not establish this containment claim.