The parking 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 $S$. A parking edge is an edge appearing in a flip sequence that is not contained in $T_{\text{in}}\cup T_{\text{tar}}$. Parking edge conjecture. There is a shortest flip sequence from $T_{\text{in}}$ to $T_{\text{tar}}$ that only uses parking edges from the boundary of the convex hull of $S$. The conjecture restricts temporary edges in shortest flip sequences to the convex-hull boundary; the supplied text states that a compatible-flip analogue is proved, but does not state that this original unrestricted-flip conjecture is resolved.