The dg-quiver Schur algebra quasi-isomorphism conjecture

Let $({}_{dg}Q_b^{{\underline{N}}},d_N)$ be the dg-quiver Schur algebra defined as the dg-endomorphism algebra of the direct sum of the dg-enhanced modules $G_{\rho}^{{\underline{N}}}$, and let $(Q_b^{{\underline{N}}},0)$ denote the corresponding ordinary algebra. Quasi-isomorphism conjecture. There is a quasi-isomorphism

$$({}_{dg}Q_b^{{\underline{N}}},d_N)\xrightarrow{\simeq}(Q_b^{{\underline{N}}},0).$$

The conjecture predicts that the dg-quiver Schur algebra is quasi-isomorphic to its non-differential counterpart; the supplied text gives no resolution status.