Köhler–Roessler relative equivariant fixed-point formula conjecture

Let $Y$ and $B$ be $\mu_n$-equivariant arithmetic varieties over an arithmetic ring $D$, and let $f:Y\to B$ be flat, $\mu_n$-projective, and smooth over the complex numbers. Write $Y_{\mu_n}$ and $B_{\mu_n}$ for the fixed loci, let $f^{\mu_n}:Y_{\mu_n}\to B_{\mu_n}$ be the induced map, and let $\lambda_{-1}$ and $R_g$ have their usual equivariant K-theoretic and equivariant $R$-genus meanings. Define

$$M(f)=\bigl(\lambda_{-1}(N_{Y/Y_{\mu_n}}^{\vee})\bigr)^{-1}\lambda_{-1}(f^{*}N_{B/B_{\mu_n}}^{\vee})\bigl(1-R_g(N_{Y/Y_{\mu_n}})+R_g(f^{*}N_{B/B_{\mu_n}})\bigr).$$

Relative fixed-point formula conjecture. The diagram

$$\begin{CD} K^{\mu_n}_{0}(Y) @>{M(f)\,\rho(\cdot)}>> K^{\mu_n}_{0}(Y_{\mu_n})\otimes_{R(\mu_n)}\mathcal R \\ @V{f_*}VV @VV{f^{\mu_n}_*}V \\ K^{\mu_n}_{0}(B) @>{\rho(\cdot)}>> K^{\mu_n}_{0}(B_{\mu_n})\otimes_{R(\mu_n)}\mathcal R \end{CD}$$

commutes. This conjecturally generalizes the fixed-point theorem of Köhler and Roessler to the relative setting; the source does not establish commutativity in general.