The Meta-Isomorphism Conjecture for spaces-to-spectra functors with coefficients

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. Let $G$ be a group, ${\mathcal F}$ a family of subgroups, and $Z$ a free $G$-CW-complex. Write ${\mathbf S}^G_Z$ for the associated equivariant coefficient spectrum, with $H_n^G(G/G;{\mathbf S}^G_Z)=\pi_n({\mathbf S}(Z/G))$. Meta-Isomorphism Conjecture with coefficients. For every free $G$-CW-complex $Z$, the assembly map induced by $\operatorname{pr}\colon E_{\mathcal F}(G)\to G/G$ is bijective for all $n\in{\mathbb Z}$:

$$H_n^G(\operatorname{pr};{\mathbf S}^G_Z)\colon H_n^G(E_{\mathcal F}(G);{\mathbf S}^G_Z)\to H_n^G(G/G;{\mathbf S}^G_Z)=\pi_n({\mathbf S}(Z/G)).$$

The freeness condition is essential according to the surrounding remark; the supplied text gives no resolution status.