In this paper, we extend the Goldman-Millson Theorem for $L_\infty$ algebras. We consider two $L_\infty$ algebras $L$ and $\tilde{ L}$ endowed with descending, bounded above and complete filtrations compatible with the $L_\infty$ structures and ${ U:L \rightarrow \tilde{ L}}$ an $\infty$-morphism respecting the filtrations. We prove that in the setting of the linear part of $U$, say $\psi$, being a quasi-isomorphism on the r-1st page of the spectral sequences and ${ H^1 ((\mathfrak{ F}_{ 2^q} L)/(\mathfrak{ F}_{ \mathrm{ min}(2^{ q+1},r)} L))=0}$ for every $q$ with $2^q < r$ and ${ H^i((\mathfrak{ F}_1 \tilde{ L}) / (\mathfrak{ F}_q \tilde{ L}))=0}$ for $i=0,1$ and $q$ every power of 2 smaller than $r$ and $q=r$ this induces a weak homotopy equivalence of the simplicial sets $\mathfrak{ MC}_\bullet (L)$ and $\mathfrak{ MC}_\bullet (\tilde{ L})$.
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