I have to deal with unlimited leaks and I want to use the conditional convergence of spectral sequences and the results of
(one): 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 pair that comes 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} $ Y $ r in mathbb {N} $ ($ r $ is the page number and $ s $ the “ degree of filtration & # 39; & # 39;). 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 on $ mathbb {R} $.
The following are the two theorems (or their parts) of (1) that interest me:
Theorem 6.1 (p.19): Leave $ 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} Yes $ A ^ infty = 0 $, then the spectral sequence
converges strongly to $ A ^ { infty} $.Theorem 7.2 (p.21): Leave $ f: C rightarrow bar {C} $ Be a morphism of filtered cochain complexes and assume that $ E ^ s $, resp. $ bar {E} ^ s $
conditionally converge to $ A ^ { infty} $, resp. $ bar {A} ^ { infty} $.
Suppose, in addition, that begin {equation} tag {C2} E ^ s = bar {E} ^ s =
0 text {for all} s <0. end {equation} Yes $ f $ induces the
isomorphisms $ E ^ infty simeq bar {E} ^ infty $ Y $ RE ^ infty simeq
R bar {E} ^ infty $, then induces isomorphism $ H (C) simeq H ( bar {C}) $.
Let me introduce the standard (grade changed) beamed in $ E_r $ and visualize $ E_r ^ s, d} $ as sitting on the coordinate $ (s, d) $ flat. The differentials are then.
$$ d_r: E_r ^ {s, d} rightarrow E_r ^ {s + r, dr + 1}. $$
This are my questions:

How is Theorem 6.1 generalized if (C1) is replaced by the following? Condition of differential outputs?
$$ E_r text {sit on a semiplane and if we fix some coordinate} (s, d), text {then almost all finely} d_r text {starting at} (s, d) text {leave the semiplane.} $$ 
How is Theorem 7.2 generalized if (C2) is replaced by the following? Condition of entering differentials?
$$ E_r text {sit on a semiplane and if we fix some coordinate} (s, d), text {then almost all finely} d_r text {ending in} (s, d) text {start outside the semiplane .} $$
The author of (1) answers the questions as follows:

On p.19, Chapter 6 in parentheses just before Theorem 6.1:
…The
the results are properly generalized, since all arguments can be carried out gradually; the
The main difficulty is finding a notation that helps instead of hindering exposure. 
On p.20, chapter 7 in parentheses, a couple of paragraphs before theorem 7.2:
… The results are still valid when properly modified, like all arguments
it can be carried out gradually; the difficulty is finding notation that
Help instead of hindering.
How do these theorems generalize precisely? Has it been done somewhere? Thank you!
PD I come from differential geometry and I am not familiar with the test methods for spectral sequences. I use it simply as a black box.