Klaus–Teichner conjecture on signature multiplicativity modulo 8

Let $F^{2m} \to E \to B^{2n}$ be a fibration of oriented Poincaré complexes, and suppose that the action of $\pi_1(B)$ on $H^{m}(F;\mathbb Z_2)$ is trivial. Klaus–Teichner's conjecture. The signature is multiplicative modulo $8$:

$$\sigma(E)-\sigma(F)\sigma(B)=0 \in \mathbb Z_8.$$

This conjecture weakens the usual trivial-action condition used in signature-multiplicativity results. The source describes an attempted proof using spectral sequences, but supplies no resolution of the conjecture.