The K4 subgraph conjecture for uniquely 3-list-colorable planar graphs

A U$3$LC planar graph is a planar graph that is uniquely $3$-list colorable. In other words, it has a $3$-list assignment from which it has a unique list coloring.

K4 subgraph conjecture. Every U$3$LC planar graph has $K_4$ as a subgraph.

The conjecture is motivated by the preceding result that a plane graph with at most seven triangular faces cannot be uniquely $3$-list colorable. No resolution is given in the supplied source.