Local cyclic index theorem conjecture

Let $A$ be a local elliptic operator. Let $T^{\lambda, loc}_{S}.\operatorname{Index}(A)$ denote its local cyclic topological index class and let $A^{\lambda, loc}.\operatorname{Index}(A)$ denote its local cyclic, $S$-localised analytical index class. Let $\partial^{\lambda, loc}$ be the connecting homomorphism in the local cyclic complex, composed with the isomorphism induced by $S$-localisation.

Local cyclic index theorem conjecture. For any local elliptic operator $A$,

$$\partial^{\lambda, loc}(T^{\lambda, loc}_{S}.\operatorname{Index})(A)=(A^{\lambda, loc}.\operatorname{Index})(A).$$

This is the cyclic-homology analogue of the preceding local $K$-theory index statement. The supplied text gives no evidence that it has been resolved.