Remaining open cases of Jeffs’ conjecture and wheel types for certain nerves

Ascertain whether Jeffs’ conjecture holds for non-minimal L24 codes and for all codes whose nerve of maximal codewords is L25, L26, L27, or L28; additionally, identify what types of wheels can occur in such codes and whether sprockets suffice or other wheel structures are needed.

Background

The authors prove Jeffs’ conjecture for the nerve cases L1–L23 and for minimal L24 codes, and show non-convexity via sprockets for certain L24/L28 examples. However, they explicitly identify non-minimal L24 codes and all L25–L28 cases as remaining open for the conjecture, and note uncertainty about which wheel types appear in these families.

Understanding wheel structures beyond sprockets could be essential for a complete obstruction-based characterization in these cases.