Clouâtre–Ostermann–Ransford conjecture

The conjecture, formulated in a 2020 paper on an abstract approach to Crouzeix’s conjecture, predicts a sharp bound of $2$ for a homomorphism under a symmetrized contractivity condition. It is an abstract formulation whose consequences include the numerical-range conjecture.

Known results

• The 2020 paper proves the weaker general bound $1+\sqrt{2}$, shows sharpness without unitality of the antilinear map, and proves the conjecture in two special cases.
• A related result claims the completely bounded variant in a special operator-algebraic formulation.

2026 claimed proof

A 2026 arXiv preprint, A solution to Crouzeix’s conjecture, claims the abstract conjecture with a bounded, rather than contractive, unital antilinear map and an operator-valued target on a Hilbert space. It also claims that every numerical range is a $2$-spectral set, but no independent verification, referee report, error report, or retraction was found.

Current status (as of September 2026): A preprint claims the abstract conjecture and its numerical-range consequence are proved, but they remain unverified; implications for other variants remain open.