at.algebraic topology - Bousfield-Kan spectral sequence with local coefficients

Let $F : mathcal{D} to mathbf{Top}$ be a diagram of topological spaces. A local system of coefficients $M$ on $mathrm{colim}_mathcal{D} F$ pulls back to a local system $M_d$ on $F(d)$ for each $d in mathcal{D}$, and also a local system $M_h$ on $mathrm{hocolim}_mathcal{D} F$.

Is there a Bousfield-Kan type spectral sequence of the form

$$E^2_{s,t} = mathrm{colim}^s_{mathcal{D}} H_t(F(d);M_d) Rightarrow H_{s+t}(mathrm{hocolim}_mathcal{D} F;M_h)$$
and if so where can one find it in the literature? I would also be content to know if this is not possible.

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