Conjecture on twisted L-groups and Q-groups from algebraic Wu classes

Let $(B_*,\beta)$ be a chain bundle with algebraic Wu class $\widehat v_{2k}(\beta)$, and let $\operatorname{ev}$ denote evaluation on homology. Let $\Sigma$, $\Sigma_8$, $\Lambda$, $\operatorname{Coker}$, $Q^-$, and $\iota_1$ be the constructions defined in the source. Conjecture on twisted groups. If $(B_*,\beta)$ is $(2k-1)$-connected, then

$$L^{4k}(B_*,\beta)\cong W_0(Q_\beta(2k))\cong\Sigma\bigl(\operatorname{ev}(\widehat v_{2k}(\beta))\bigr)$$

and

$$Q_{4k}(B_*,\beta)\cong\Sigma_8\bigl(\operatorname{ev}(\widehat v_{2k}(\beta))\bigr).$$

If $(B_*,\beta)$ is $(2k-2)$-connected and $\operatorname{ev}(\widehat v_{2k}(\beta))=0$, then $[v'_{Q_\beta(2k-1)}]=\widehat v_{2k}(\beta)$ and

$$L^{4k-2}(B_*,\beta)\cong W_0(Q_\beta(2k-1))\cong\Lambda(v'_{Q_\beta(2k-1)})$$

and

$$Q_{4k-2}(B_*,\beta)\cong\operatorname{Coker}\bigl(\Lambda(\iota_1)\to\Lambda(v'_{Q_\beta(2k-1)})\bigr).$$

If instead $(B_*,\beta)$ is $(2k-2)$-connected and $\operatorname{ev}(\widehat v_{2k}(\beta))\neq0$, then $Q_\beta(2k-1)\cong Q^-\oplus H_{2k-1}(B_*)$ and

$$L^{4k-2}(B_*,\beta)\cong Q_{4k-2}(B_*,\beta)\cong W_0(Q_\beta(2k-1))\cong\Lambda(H_{2k-1}(B_*))$$

The conjecture is presented as a consequence of the preceding conjecture, the proposition computing quasi-Wu classes, and the theorem computing Witt groups; the source supplies no resolution.