Determine whether the 28-vertex K4-free-complement problem is computationally feasible

Determine whether the enumeration-based method using exact unions of two sphere triangulations and SAT certification is feasible for finding a biplanar graph on 28 vertices with $K_4$-free complement.

Background

Gethner and Sulanke’s Open Problem 1(b) asks for a biplanar graph on 28 vertices whose complement is $K_4$-free. The paper explains that its enlargement and exact-union encoding extend in principle to this order, but the required enumeration would be much larger. The authors explicitly leave unresolved whether the computational approach is feasible at that scale.