Local K-theory connecting homomorphism isomorphism conjecture

Let $M$ be the underlying compact manifold, let $\Psi^{0}(M)$ and $\Psi^{-1}(M)$ denote the algebras of operators of orders $0$ and $-1$, respectively, and let $K_{i}^{loc}$ denote local $K$-theory. The quotient $\Psi^{0}(M)/\Psi^{-1}(M)$ has connecting homomorphism

$$\partial^{K, loc}: K_{1}^{loc}(\Psi^{0}(M)/\Psi^{-1}(M)) \longrightarrow K_{0}^{loc}(\Psi^{-1}(M)).$$

Local K-theory connecting homomorphism conjecture. In local $K$-theory, $\partial^{K, loc}$ is an isomorphism rationally.

This conjecture asserts the expected rational exactness phenomenon for the local $K$-theory sequence associated with the order filtration of pseudodifferential operators. The supplied text gives no evidence that it has been resolved.