Real equivariant conjectures for truncated Brown–Peterson spectra

Work at $p=2$. Let $\sigma$ be the sign representation, $\rho=\sigma+1$ the regular representation, and let $T(n)_\mathbf{R}$ and $X(2^n)_\mathbf{R}$ be the Real spectra defined from the equivariant Quillen summand. Let $\mathrm{BP}\langle n\rangle_\mathbf{R}$ be the Real truncated Brown–Peterson spectrum, and use the relative Real topological Hochschild and periodic homology constructions in the statement.

Real equivariant conjectures. The six listed assertions in the source hold: the required $\mathbf{E}_{\rho}\rtimes\mathrm{U}(1)_\mathbf{R}$-algebra and splitting structures exist; the two displayed relative $\mathrm{THH}_\mathbf{R}$ equivalences and detection statement hold; the equivariant spaces $\widetilde K_n$ and their fibrations have the stated properties; the equivariant topological Sen cofiber sequence exists; the asserted parametrized-Tate equivalence for $\mathrm{TP}_\mathbf{R}$ holds; and, for a $2$-complete animated commutative ring with trivial $\mathbf{Z}/2$-action, the stated filtration with graded pieces (\hat{{\mathbbl{\Delta}}}_{R/\mathbf{Z}[\\![q-1]\\!]}^\wedge_2)[2i] exists. These claims extend the paper's constructions to Real/equivariant settings, but no resolution is supplied.