Weak Haagerup constant multiplicativity conjecture

The conjecture asks whether the weak Haagerup constant of a direct product always equals the product of the two constants.

Known results

• The inequality $\Lambda_{\mathrm{WH}}(G\times H)\leq\Lambda_{\mathrm{WH}}(G)\Lambda_{\mathrm{WH}}(H)$ was proved for locally compact groups in 2014; equality was noted as open, except when either factor has constant $1$ (The weak Haagerup property, 2014).

September 28, 2026 preprint claim

A new preprint by Cédric Arhancet claims the equality $\Lambda_{\mathrm{WH}}(G\times H)=\Lambda_{\mathrm{WH}}(G)\Lambda_{\mathrm{WH}}(H)$ for all locally compact groups. The claim is based on an unrefereed preprint and has not been independently verified.

Current status (as of September 2026): The conjecture is claimed solved for all locally compact groups, but the claimed proof remains unverified.