The weak happy edge conjecture for plane spanning trees

Let $S$ be a convex point set, and let $T_{\text{in}}$ and $T_{\text{tar}}$ be plane spanning trees on $S$. An edge is happy if it belongs to both $T_{\text{in}}$ and $T_{\text{tar}}$. Weak happy edge conjecture. There is a shortest flip sequence from $T_{\text{in}}$ to $T_{\text{tar}}$ that does not flip happy edges. This conjecture asks whether shortest reconfiguration sequences can preserve all edges common to the initial and target trees. Its resolution is not given in the supplied text.