Unitary-rank analogue of the modified Xiao conjecture

Let $f\colon S\to B$ be a fibred surface, meaning a fibration from a smooth projective surface $S$ to a smooth projective curve $B$, with general fibre of genus $g\geq 2$. Let $u_f$ denote the rank of the flat unitary summand of the Hodge bundle, and call that summand finite-monodromy when its monodromy representation has finite image. Call $f$ non-isotrivial when its smooth fibres are not all mutually isomorphic. Unitary-rank analogue of the modified Xiao conjecture. For every non-isotrivial fibred surface whose flat unitary summand has finite monodromy, one has

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

The inequality follows in the finite-monodromy case by a base change that trivializes the flat summand and then applying the relative-irregularity bound. The paper contrasts this conditional statement with examples of Catanese and Dettweiler having infinite monodromy; the supplied status is unknown, so the claim remains open in the database.