It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My question is:under which conditions does this spectral sequence degenerate at $E_1$ (actually, I am more interested in necessary conditions)? This condition appears to be equivalent to the image of the morphism $f_1:E_1\to D_1$ lying inside all levels of the filtration of $D_1$ by $g_1^i(D_1)$ (here I ignore the upper indices, and $g_1^i$ denotes the $i$th iterate of $g_1:D_1\to D_1$). Is this true; are there any references for this fact?

Actually, I would like to conclude that $f_1=0$; does this follow from the degeneration at $E_1$? I am more interested in the bounded case; so an answer to the first part of my question is sufficient for my purposes; yet are there any other conditions ensuring that $f_1=0$ whenever the spectral sequence degenerates at $E_1$?