Non-triviality conjecture for exotic cyclic and Hochschild classes on the two-torus

Let $C^{2/3}(S^1\times S^1)$ denote the Hölder algebra of exponent $2/3$ on the two-torus. Let $\mathfrak{c}_\omega$ be the cyclic cochain and $\mathfrak{h}_\omega$ its associated Hochschild cochain, constructed from the Fredholm module of the cited theorem. Their cohomology classes lie in

$$[\mathfrak{c}_\omega]\in HC^{2}_\lambda(C^{2/3}(S^1\times S^1)),\qquad [\mathfrak{h}_\omega]\in HH^{3}(C^{2/3}(S^1\times S^1)).$$

Non-triviality conjecture. The cyclic cohomology class $[\mathfrak{c}_\omega]$ and the associated Hochschild cohomology class $[\mathfrak{h}_\omega]$ are non-trivial. These classes are proposed as exotic cohomology classes arising from a Fredholm module at the critical Hölder regularity; their non-triviality is not established in the supplied text.