Does any insulator host a spinon Fermi surface?
In plain words
A spinon Fermi surface would make an electrical insulator behave like a metal for spin and heat. Candidates include organic triangular-lattice salts and the layered compound 1T-TaS2, but no measurement has established it.
Precise statement
Identify an insulating magnet in a $U(1)$ spin-liquid state with a spinon Fermi surface, signaled by a magnetic specific heat $C \sim T^{2/3}$ (or $C/T \to \mathrm{const}$ if gauge fluctuations are weak) and a magnetic $\kappa/T \to \mathrm{const}$ as $T \to 0$, plus $2k_F$ singularities in the spin structure factor or quantum oscillations of a neutral Fermi surface. An answer is a material with at least two independent such signatures reproduced across samples, or a demonstration that all present candidates have other ground states.
What would settle it
Combined low-temperature heat capacity, thermal conductivity and spectroscopic evidence on one material, reproduced by independent groups.
Status in the literature
A residual $\kappa_{0}/T > 0$ in EtMe3Sb[Pd(dmit)2]2 (Yamashita et al., Science 2010) was not reproduced by Ni et al. (PRL 2019) or Bourgeois-Hope et al. (PRX 2019); no candidate has two reproduced signatures.