The Tor description conjecture for cohomology of Witt groups

Let $F$ be a field with $|\cdot F/\cdot F^2|=2^n$, so that $E=E_n\cong (\mathbb Z/2)^n$, and let ${\mathcal G}_F$ be the associated $W$-group. Let $\kappa_1,\dots,\kappa_r\in H^2(E_n)$ be the $k$-invariants of the central extension defining ${\mathcal G}_F$. Choose degree-two polynomial generators $b_1,\dots,b_r$ with $b_i\mapsto\kappa_i$, and let $\zeta_1,\dots,\zeta_r$ be the corresponding permanent cocycles in bidegree $(-1,3)$. The Tor description conjecture. Up to filtration, there is an isomorphism of algebras

$$H^*({\mathcal G}_F)/(\zeta_1,\dots,\zeta_r)\cong \operatorname{Tor}_{\mathbb F_2[b_1,\dots,b_r]}(\mathbb F_2[\dot F/\dot F^2],\mathbb F_2).$$

This conjecture describes the cohomology of a $W$-group in terms of the Eilenberg–Moore spectral sequence associated with its central extension. The source states that it has been verified for all examples known to the authors, but does not establish it in general.