The compatibility of the cyclotomic trace with p-adic Adams operations

Let $R$ be a connective $E_{\infty}$ ring spectrum, and let

$$K(R)^{\scriptscriptstyle\wedge}_{p}\longrightarrow TC(R)^{\scriptscriptstyle\wedge}_{p}$$

be the cyclotomic trace, where both spectra are $p$-completed. The cyclotomic-trace compatibility conjecture. The cyclotomic trace commutes with the conjectural $p$-adically interpolated Adams operations. This compatibility is proposed as a second-level conjecture because it would help relate the Adams operations on topological cyclic homology to those conjectured on algebraic $K$-theory and thereby explain the splittings in the paper; the supplied text gives no resolution.