Conjecture on the map from W_0(Q_beta(q)) to twisted L-groups

Let $(B_*,\beta)$ be a chain bundle, let $q\geq0$, and let $\eta_\beta\colon W_0(Q_\beta(q))\to L^{2q}(B_*,\beta)$ be the natural homomorphism. Conjecture on $\eta_\beta$. If $H_{q-1}(B_*)$ is torsion free, then $\eta_\beta$ is split injective. If, in addition, $H_i(B_*)=0$ for all $i<q$, then $\eta_\beta$ is an isomorphism. The conjecture would identify the Witt group of the form parameter with the relevant twisted $L$-group under the stated connectivity and torsion hypotheses; the source gives a proposed construction of an inverse but no proof.