Novikov-homology criterion for fibred links

Let $L$ be a link, let $C_L$ be its exterior, and let $\xi:\pi_1C_L\to\mathbf{Z}$ be the homomorphism sending each meridian generator to $1$. Let $\Lambda$ be the group ring of $\pi_1C_L$, let $\widehat{\Lambda}_\xi$ be the corresponding Novikov ring, let $\widetilde{C_L}$ be the universal cover, and let $C_*^\Delta(\widetilde{C_L})$ be its cellular chain complex. The Novikov-homology fibredness conjecture. A link $L$ is fibred if and only if

$$H_*\left(\widehat{\Lambda}_\xi\underset{\Lambda}{\otimes}C_*^\Delta(\widetilde{C_L})\right)=0.$$

The vanishing is necessary for fibred links, but the text presents sufficiency as a conjectural criterion; the supplied passage does not give a resolution of this assertion.