The predicted splitting of the hyperreal dual Steenrod algebra

Let $G$ be the group acting in the hyperreal setting, let $C_2$ be its indicated subgroup, and let $E_{2+\sigma}$ and $BBU_{\mathbb R}$ denote the spaces appearing in the source. Write $H\underline{\mathbb Z}$ for the relevant equivariant Eilenberg–Mac Lane algebra and let $\Xi_n$ denote the quotient object introduced earlier in the paper. The group $G$ acts on $\operatorname{Map}^{C_2}(G,E_{2+\sigma})$ and $\operatorname{Map}^{C_2}(G,BBU_{\mathbb R})$. Predicted splitting. There should be an equivalence of $\operatorname{Map}^{C_2}(G,E_{2+\sigma})$-algebras

$$H\underline{\mathbb Z}\underset{\Xi_n}{\otimes}H\underline{\mathbb Z}\simeq H\underline{\mathbb Z}\otimes\Sigma^{\infty}_+\operatorname{Map}^{C_2}(G,BBU_{\mathbb R}).$$

The preceding geometric-fixed-point computation suggests this splitting by identifying the left-hand side with the expected additional copy of $BBU_{\mathbb R}$; establishing the asserted equivalence requires proving that this geometric-fixed-point evidence lifts to an equivalence of algebras.