Universal Segre-SM classes for multi-singularity constructible functions

For a stable map $f:M\to N$, let $M(\underline{\eta})(f)$ be the closure of the locus of tuples of distinct points mapping to a common point, with the map-germ at each point having the prescribed stable multi-singularity type $\underline{\eta}$. Let $\alpha_{\underline{\eta}}$ and $\beta_{\underline{\eta}}$ be the corresponding source and target multi-singularity constructible functions, and let $c_i=c_i(f)$ denote the quotient Chern classes and $s_I$ the Landweber–Novikov classes. Universal Segre-SM class conjecture. For any stable multi-singularity type $\underline{\eta}$ in relative codimension $\kappa$, there exist power series $tp^{{\rm SM}}(\alpha_{\underline{\eta}})$ and $tp^{{\rm SM}}(\beta_{\underline{\eta}})$ in the classes $c_i$ and $s_I$ such that, for every stable map $f:M\to N$ of relative codimension $\kappa$,

$$tp^{{\rm SM}}(\alpha_{\underline{\eta}})=c(TM)^{-1}C_*(\alpha_{\underline{\eta}}), \qquad tp^{{\rm SM}}(\beta_{\underline{\eta}})=c(TN)^{-1}C_*(\beta_{\underline{\eta}})$$

in $H^*(M)$ and $H^*(N)$, respectively. This would simultaneously generalize the universal Segre-SM classes for mono-singularities and the Thom polynomials of multi-singularities; the cited low-dimensional results support the conjecture, but its general status is unresolved.