The question is pretty much in the title. It is a classical fact that a filtered dga gives rise to a multiplicative spectral sequence. It is claimed in Remark 4.1 of https://arxiv.org/pdf/1410.6728.pdf that this generalizes to filtered $A_\infty$- algebras(see the page 27 of the same for this definition). One can define a notion of filtered $A_N$-algebras for $N \in \mathbb{N}^{\geq 0}$, $N \geq 3$ where $m_i$ are only defined for $i \leq N$ and the $A_{\infty}$ equations hold for $i \leq N$ (so the multiplication is still associative on the level of homology).

Question: Is the natural spectral sequence associated to a filtered $A_N$ algebra multiplicative?

I would guess the answer is yes, but could not find a reference for this fact and there seems to be some subtleties in the literature according to this earlier post:

Multiplicative structure on spectral sequence

Motivation: It seems like this might be an efficient way to prove that some spectral sequences that arise "in nature" are multiplicative, say by equipping the relevant (co) chain complex with an $A_3$ structure.