Upgrade the S1-equivariant E1-equivalence to an E∞-equivalence

Prove that the S1-equivariant equivalence of E1-ring spectra THH(ℤp[ζp]/S)ˆp ≃ τ≥0(ku^{tCp}) can be promoted to an S1-equivariant E∞-equivalence of ring spectra.

Background

The authors use an S1-equivariant E1-equivalence between THH(ℤp[ζp]/S)ˆp and τ≥0(ku^{tCp}) as a key input to relate refined TC− and q-de Rham theory. They remark that an E∞-level refinement is conjectured to hold.

Establishing the E∞-structure would strengthen the foundations of the comparison and could facilitate further refinements and functoriality at the cyclotomic and multiplicative levels.