Modified Xiao's conjecture for the irregularity of fibrations

Let $f:S\rightarrow B$ be a fibration from a compact surface $S$ to a compact curve $B$, meaning a surjective morphism with connected fibres, and let $F$ be a general smooth fibre. Write $g$ for the genus of $F$ and define the relative irregularity by $q_f=q(S)-g(B)$. The fibration is non-trivial if $S$ is not birational to $B\times F$ with the given fibration induced by the first projection.

Modified Xiao's conjecture. Every non-trivial fibration satisfies

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

or equivalently

$$q_f\leq\left\lceil\frac{g+1}{2}\right\rceil.$$

This modification is motivated by the fact that the known counterexamples to Xiao's original conjecture fail its bound by exactly $\frac12$. The source records a result of Pirola proving the modified conjecture under a constancy condition on an associated Abel--Jacobi map, but the displayed status evidence identifies the conjecture as disproved.