Existence conjecture for Kodaira fibrations without topological sections

A topological section of a Kodaira fibration $f:S\to C$ is a splitting of its short exact sequence of fundamental groups.

Existence conjecture. There is a Kodaira fibration with no topological section. The source motivates this by the universal family of curves, which has no topological sections, but states that no such Kodaira fibration is currently known.