Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, suppose $E_N=E_{N+1}=\cdots=E_{\infty}$ for some fixed number $N$. I have the following questions:

- Can we construct a filtered chain complex $(C,d)$ such that the induced spectral sequence is $\{(E_r,d_r)\}$?
- Can we determine $(C,d)$ up to some equivalence (like filtered quasi-isomorphism)?
- Can we relax the assumption to obtain the similar results?
- Are there any good references?

Indeed, the spectral sequence in my mind is from the unrolled spectral sequence in

https://ncatlab.org/nlab/show/conditional+convergence

There are many similar questions, which gives some potential counterexamples when we replace $\mathbb{F}$ by $\mathbb{Z}$ or without the assumption about the convergence at some finite page $E_N$.

Complete invariant of filtered chain complexes under chain homotopy equivalence

Functoriality of filtered spectral sequences

isomorphic spectral sequences => quasi-isomorphic filtered chain complexes?