Xiao's conjecture on the genus of canonically fibered surfaces

Let $f:X\dashrightarrow C$ be a canonically fibered surface, meaning that $X$ and $C$ are irreducible, smooth and projective over $\mathbb{C}$ with dimensions $2$ and $1$, respectively, $f$ has connected fibers, $X$ is of general type, and

$$|K_X|=f^*\mathcal D+\mathcal N$$

for a base point free linear series $\mathcal D$ on $C$ of dimension at least $1$ and fixed part $\mathcal N$. Xiao's conjecture. The fiber

$$X_p=f^{-1}(p)$$

is a curve of genus at most $4$ for a general point $p\in C$ when $p_g(X)\gg 1$. This conjecture predicts a sharp upper bound on the genus of the general fiber of a canonically fibered surface in the large-geometric-genus regime. The paper addresses gaps in the relative-genus-$5$ case and applies related arguments to relative genera $3$ and $4$; the supplied text does not establish the conjecture itself.