Voiculescu’s cyclic-gradient conjecture

Prove that conjugate variables can always be approximated in L² by cyclic gradients, at least for Lipschitz conjugate variables.

Background

The paper notes that the distinction between free Gibbs laws and general tuples with Lipschitz conjugate variables would be reduced if conjugate variables were approximable by cyclic gradients. This is identified as Voiculescu’s cyclic-gradient conjecture, with the authors noting that a positive answer is believed likely in the Lipschitz setting.

References

The picture for free Gibbs laws and that for Lipschitz conjugate variables would, however, be much closer if Voiculescu's cyclic-gradient conjecture were true: this conjecture asks whether conjugate variables can always be approximated in $L2$ by cyclic gradients (see Section~0.2, Q.4 for precise details).

— Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound  (2610.22109 - Diez, 15 Oct 2026) in Section 1, subsection “Positive free curvature and the spectral Bochner argument”