The mod-8 signature conjecture for 2-adic Poincare complexes

Let $Z$ be a $2$-adic Poincaré simply-connected finite CW complex of dimension $4n$, with mod-$2$ Spivak normal spherical fibration

$$\nu_Z:Z\rightarrow BSG_2.$$

The Poincaré duality pairing on $H^{2n}(Z;\mathbb{Z}/8)$ gives a nonsingular symmetric bilinear form.

Mod-8 signature conjecture. The evaluation

$$\langle l^G(\nu_Z),[Z]_2\rangle\in\mathbb{Z}/8$$

is the mod-$8$ signature of the symmetric bilinear form on $H^{2n}(Z;\mathbb{Z}/8)$ induced by Poincaré duality.

This conjecture extends the known identification of the corresponding characteristic-class evaluation with the signature modulo $8$ for finite CW complexes with integral Poincaré duality. The source does not provide evidence that the conjecture has been resolved.