Eaton–Moretó conjecture

The Eaton–Moretó conjecture predicts an inequality of the form $mh(B)\leq mh(D)$ for a block $B$ with defect group $D$. Recent work extends the known range substantially but does not address arbitrary blocks of arbitrary finite groups.

Known results

• 2014: Established for several families, including principal blocks of quasi-simple groups and specified Lie-type, symmetric, alternating, and covering groups.
• September 2024: Proved for principal blocks of finite $p$-solvable groups with nonabelian defect groups.
• October 2024: Ruled out minimal counterexamples among quasi-simple groups and obtained further results for $p\geq 5$ and selected $p=2,3$ cases.

August 2026 claimed extensions

A paper dated August 2026 claims the conjecture for all $p$-solvable groups. A separate paper by Navarro and Gómez–Serrano claims the principal-block inequality for arbitrary finite groups and, assuming Dade’s Projective Conjecture, the full principal-block case. Its proof used ChatGPT-5.6-Sol and Claude Fable 5, but the final argument was written by the authors; neither claim has independent verification.

Current status (as of August 2026): A proof is claimed for all $p$-solvable groups and for principal blocks of arbitrary finite groups, but these claims are unverified; the conjecture for arbitrary finite groups and nonprincipal blocks remains open.