Equivariant model conjecture for finite regular covers

Let $X$ be a connected CW-complex, and let $Y\to X$ be a finite regular cover with deck group $\Phi$. Suppose that $Y$ has finite Betti numbers. Let $A$ be a $\Phi$-commutative differential graded algebra, and assume there is a zigzag of quasi-isomorphisms connecting the polynomial forms model

$$A_{\scriptscriptstyle{{\rm PL}}}(Y)\otimes_{\mathbb{Q}}\mathbb{k}$$

to $A$ in commutative differential graded algebras, such that the induced isomorphism

$$H^*(Y,\mathbb{k})\cong H^*(A)$$

is $\Phi$-equivariant.

Equivariant model conjecture. The fixed subalgebra $A^{\Phi}$ is a model for $X$.

This conjecture seeks to recover a model for an orbit space from an equivariant model of its finite cover. The source gives no resolution status; the preceding proposition shows a related assertion under the stronger hypothesis of a zigzag of equivariant $q$-equivalences.