The quasi-isomorphism conjecture for dg-cyclic modules

Let $G_{\rho}^{{\underline{N}}}$ and $Y_{\rho}^{{\underline{N}}}$ be the dg-enhanced and ordinary cyclic modules constructed from the tensor-product data, with differential $d_N$ on $G_{\rho}^{{\underline{N}}}$. Quasi-isomorphism conjecture. There is a quasi-isomorphism

$$(G_{\rho}^{{\underline{N}}},d_N)\xrightarrow{\simeq}(Y_{\rho}^{{\underline{N}}},0).$$

This conjecture asserts that the differential enhancement recovers the corresponding ordinary cyclic module up to quasi-isomorphism; the supplied text gives no resolution status.