Non-commutative correspondence conjecture for derived $\Gamma$-schemes

Let $X_{\Gamma}^{\mathrm{nc}}$ be a compact non-commutative derived $\Gamma$-scheme, and let $\mathcal C_\Gamma^{\mathrm{dg}}$ be the dg-category of $\Gamma$-representations. Non-commutative correspondence conjecture. Compact non-commutative derived $\Gamma$-schemes correspond contravariantly to dg-categories of $\Gamma$-representations:

$$X_{\Gamma}^{\mathrm{nc}}\simeq\mathcal C_\Gamma^{\mathrm{dg}}.$$

The proposed correspondence is part of the paper's non-commutative extension of derived $\Gamma$-geometry; no resolution is supplied in the source.