I am trying to understand strong convergence for whole-plane spectral sequences in the paper by J.Boardman: https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf

I am currently confused about the convergence of the zig-zag double complex $D$ mentioned here as one of the answers: https://math.stackexchange.com/questions/1368777/double-complex-with-exact-rows

Firstly Let $C$ be the total complex of $D$. By Theorem 10.1 in Boardman's paper the spectral sequences $E^{\alpha}_2=H(H(D,\delta),d)$ and $E^{\beta}_2=H(H(D,d),\delta)$ converges conditionally to the cohomology $H(\widehat{C})$ where $\widehat{C}$ is the completion of $C$. Furthermore it is clear that $E^{\alpha}$ collapses at page 1 and $E^{\beta}$ collapses at page 2 - so by the remark after Theorem 7.1 we see that $RE_{\infty}=0$ for both spectral sequences. The fact that both spectral sequences collapse imply that the condition for Theorem 8.1 is also satisfied - which means that both of them have $W=0$. Theorem 8.2 then implies that both spectral sequences converge strongly to $H(\widehat{C})$.

Is this a contradiction? Because $E^{\alpha}$ implies that all the filter quotients of $H^n(\widehat{C})$ is zero for all $n$, whereas $E^{\beta}$ tells that that the filter quotients of $H^n(\widehat{C})$ are non-zero.

Thanks!