Loday's proposal for contravariant algebraic K-theory

Let $A$ be a differential graded algebra. Suppose there are $K$-contravariant groups $K^n(A)$ and a Chern character

$$\mathrm{ch}^*:K^n(A)\to HC^n(A).$$

Loday's proposal. These data should satisfy

$$\langle\pi,\xi\rangle_K=\langle\mathrm{ch}^*(\pi),\mathrm{ch}_*(\xi)\rangle_{\mathrm{cyc}},$$

where $\langle-,-\rangle_K$ denotes a prospective pairing in $K$-theory.

The proposal seeks a contravariant version of algebraic $K$-theory related to cyclic cohomology through a Chern character and compatible pairings. The source does not provide evidence resolving the proposal.