Vanishing conjecture for the fundamental cyclic class below the half-Hölder threshold

Let $C^\alpha(S^1)$ denote the Hölder algebra of exponent $\alpha$, and let $z$ be the coordinate function on $S^1$. The class $[z\otimes z^{-1}]$ is considered in reduced cyclic homology $HC_1^{\rm red}(C^\alpha(S^1))$. Vanishing conjecture. For every $\alpha\in(0,1/2)$,

$$[z\otimes z^{-1}]=0$$

in $HC_1^{\rm red}(C^\alpha(S^1))$. This would establish a cutoff complementary to the non-vanishing result for $\alpha>1/2$; the intermediate range and the precise mechanism behind the threshold are the subject of the surrounding discussion.