I have to deal with unbounded filtrations and want to use the conditional convergence of spectral sequences and the results from

[1]: *J. Michael Boardman, Conditionally Convergent Spectral Sequences, March 1999 (http://hopf.math.purdue.edu/Boardman/ccspseq.pdf)*

The article uses cohomological spectral sequences derived from the exact couple coming from a cochain complex $C$ and a decreasing filtration $F$ of $C$. The system of inclusions is $$A^s := H(F_s C) \leftarrow A^{s+1}$$ and the pages are denoted by $E^s_r$ for $s\in \mathbb{Z}$ and $r\in \mathbb{N}$ ($r$ is the page number and $s$ the ``filtration degree''). The symbol $A^\infty$ denotes the limit and the symbol $A^{-\infty}$ the colimit. The symbol $RA^\infty$ denotes the right derived module of the limit. I basically work over $\mathbb{R}$.

The following are the two theorems (or their parts) from [1] which I am interested in:

Theorem 6.1 (p.19):Let $C$ be a filtered cochain complex. Suppose that \begin{equation}\label{Eq:Exit}\tag{C1} E^s = 0\quad\text{for all } s>0.\end{equation} If $A^\infty = 0$, then the spectral sequence converges strongly to $A^{-\infty}$.

Theorem 7.2 (p.21):Let $f: C \rightarrow \bar{C}$ be a morphism of filtered cochain complexes and suppose that $E^s$, resp. $\bar{E}^s$ converge conditionally to $A^{-\infty}$, resp. $\bar{A}^{-\infty}$. Suppose, moreover, that \begin{equation}\tag{C2} E^s = \bar{E}^s = 0\quad\text{for all }s<0. \end{equation} If $f$ induces the isomorphisms $E^\infty\simeq \bar{E}^\infty$ and $RE^\infty\simeq R\bar{E}^\infty$, then it induces the isomorphism $H(C)\simeq H(\bar{C})$.

Let me introduce the standard (degree shifted) bigrading on $E_r$ and visualize $E_r^{s,d}$ as sitting at the coordinate $(s,d)$ in plane. The differentials are then $$ d_r: E_r^{s,d}\rightarrow E_r^{s+r,d-r+1}. $$ My questions are the following:

How does Theorem 6.1 generalize if (C1) is replaced by the following

**condition of exiting differentials?**$$ E_r \text{ sit in a half-plane and if we fix any coordinate }(s,d), \text{ then all but finitely many }d_r\text{ starting at }(s,d)\text{ leave the half-plane.}$$How does Theorem 7.2 generalize if (C2) is replaced by the following

**condition of entering differentials?**$$ E_r \text{ sit in a half-plane and if we fix any coordinate }(s,d), \text{ then all but finitely many }d_r\text{ ending at }(s,d)\text{ start outside of the half-plane.}$$

The author of [1] addresses the questions as follows:

On p.19, Chapter 6 in brackets right before Theorem 6.1:

...The results generalize appropriately, as all arguments can be carried out degreewise; the main difficulty is to find notation that would help rather than hinder the exposition

On p.20, Chapter 7 in brackets a couple of paragraphs before Theorem 7.2:

...The results remain valid when appropriately modified, as all arguments can be carried out degreewise; the difficulty is to find notation that helps rather than hinders.

How do these theorems generalize precisely? Has it been done anywhere? Thanks!

P.S. I come from differential geometry and am not familiar with the proof methods for spectral sequences at all. I use it merely as a black box.