Strict cyclotomic Deligne conjecture

For every natural number $n$, let $\mathcal{O}^{\cap}(n)$ be the cylinder-operad space, let $THC(A)$ and $THH(A)$ be the associated spectra, and let $\phi_m$ be the cyclotomic structure map. Let $\phi_m^{\cap}:\mathcal{O}^{\cap}(n)\to(\mathcal{O}^{\cap}(nm))^{C_p}$ be the $S^1$-equivariant homeomorphism induced by taking the $m$-fold unbranched cover of the cylinder and conformally rescaling it.

Strict cyclotomic Deligne conjecture. For every natural number $n$, the following diagram of genuine $S^1$-spectra commutes:

\begin{tikzcd} \Sigma^\infty_+\mathcal{O}^{\cap}(n)\wedge THC(A)^{\wedge n}\wedge THH(A) \ar[r] \ar[d, "\Sigma^\infty_+\phi^{\cap}_m \wedge \Delta_m \wedge \phi_m"]& THH(A)\ar[d, "\phi_m"] \\ (\Sigma^\infty_+\mathcal{O}^{\cap}(nm)\wedge THC(A)^{\wedge nm}\wedge THH(A))^{\Phi C_m} \ar[r]& THH(A)^{\Phi C_m}. \end{tikzcd}

Here the $S^1$-action on $THC(A)^{\wedge n}$ is trivial, and the left vertical map uses the lax monoidal structure map for geometric fixed points. The statement is the strict genuine-equivariant version; weaker variants replace geometric fixed points by Tate fixed points and require commutativity only up to coherent homotopy. The source presents it conditionally on the preceding colored-operad statement.