Weak topological section conjecture for Kodaira fibrations

Let $f:S\to C$ be a Kodaira fibration, with fundamental-group sequence

$$1\to \pi_1(S_b)\to \pi_1(S)\to \pi_1(C)\to 1.$$

A topological section is a splitting of this short exact sequence.

Weak topological section conjecture. The fibration $f$ admits an algebraic section if and only if its short exact sequence of fundamental groups splits. This is introduced as a conjectural criterion intended to reduce surjectivity of the topological section map; the source does not establish it.