Grünbaum's conjecture on spanning trees and co-trees of planar graphs

Let $G$ be a planar 3-connected graph. A 3-tree is a spanning tree of $G$ with maximum degree at most $3$. For a spanning tree $T$ of $G$, let $G^*=(V^*,E^*)$ be the planar dual and define its co-tree by

$$\neg T^*:= (V^*,(E(G)-E(T))^*).$$

Grünbaum's conjecture. Every planar 3-connected graph $G$ contains a 3-tree $T$ whose co-tree $\neg T^*$ is also a 3-tree.

The conjecture strengthens the separate existence of bounded-degree spanning trees by requiring the spanning tree and its dual co-tree simultaneously to have maximum degree at most $3$. It remains open; Biedl proved the weaker bound that there is a spanning tree such that both it and its co-tree have maximum degree at most $5$.