Reduced-fiber-free slope equality conjecture for fibered surfaces

Let $D$ be an effective divisor on $\overline{\mathcal{M}}_{g}$ as in the paper's definition, and let $f\colon S\to B$ be a relatively minimal fibered surface of genus $g$ whose generic fiber is not contained in $D$. Write $K_f$ and $\chi_f$ for the relative canonical divisor and relative Euler characteristic, and let $\operatorname{Ind}_D(f^{-1}(p))$ denote the Horikawa index of the fiber germ over $p\in B$.

Reduced-fiber-free slope equality conjecture. The slope equality

$$K_f^2=(12-s_D)\chi_f+\sum_{p\in B}\operatorname{Ind}_D(f^{-1}(p))$$

holds without the assumption that any fiber of the relative canonical model of $f$ is reduced.

The theorem is known under the reducedness hypothesis and in several cases without it, including the cases described immediately before the conjecture. The general removal of the reducedness assumption remains open.