The higher-outerplanarity obstruction conjecture for completely independent spanning trees

A triangulated disc is a plane graph whose bounded faces are triangles, and a graph is $k$-outerplanar if it has a planar embedding whose vertices become empty after repeatedly removing the vertices on the outer face $k$ times. A completely independent spanning tree (CIST) is a spanning tree with the pairwise independence property described in the paper.

Higher-outerplanarity obstruction conjecture. For any $k\geq 4$, there exists a 3-connected $k$-outerplanar triangulated disc with no completely independent spanning trees.

The paper constructs such an example for $k=4$, showing the first case of the proposed obstruction; the assertion for all $k\geq4$ remains open in the supplied text.