Characteristic long exact sequence conjecture for bordism groups

Let $(\xi, \mathcal P)$ be a pair consisting of a tangential structure and a characteristic class as in the definition of a characteristic pair, and let $\xi'$ be a tangential structure for a submanifold $F$. Write $\mathit{MTChar}(\xi,\mathcal P)$ for the corresponding characteristic bordism spectrum, and $\Omega_k^{(\xi,\mathcal P)}$ and $\Omega_{k-n}^{\xi'}$ for the associated bordism groups. Characteristic long exact sequence conjecture. There is a map of spectra

$$\mathcal R\colon\mathit{MTChar}(\xi,\mathcal P)\to \Sigma^n \mathit{MT\xi}'$$

such that the map $\mathcal R$ induces on $\pi_k$ the map

$$R\colon\Omega_k^{(\xi,\mathcal P)}\to\Omega_{k-n}^{\xi'}$$

sending a characteristic pair $(M,F)$ to $F$. This is intended as the homotopical map underlying a characteristic long exact sequence of bordism groups, analogous to the Smith long exact sequence; the source notes useful algebraic corollaries but does not establish the conjecture.