Xiao's conjecture on the relative irregularity of fibred surfaces

Let $f:S\rightarrow B$ be a non-trivial fibration from a smooth projective surface $S$ to a smooth projective curve $B$, with fibres of genus $g$. Write $q_f=q(S)-g(B)$ for its relative irregularity. Xiao's conjecture. One has

$$q_f\leq \frac{g}{2}+1.$$

The conjecture was motivated by the study of supporting divisors of fibred surfaces. It is refuted: Albano and Pirola produced further counterexamples to the original conjecture, with different values of $q_f$ and $g$, while still satisfying $q_f=\frac{g}{2}+1$; the source describes the displayed statement as a modified version of Xiao's conjecture.