Let $\{X^\bullet_s\}$ be a cosimplicial tower of spaces. In other words, for each fixed "tower degree" n, we have a (let's assume Reedy fibrant) cosimplicial space $X^\bullet_n$, and for each fixed cosimplicial degree $k$, we have a tower of spaces $\{X^k_s\}_s$. Suppose that for each $\ast$, the tower of (let's assume abelian) groups $\{\pi_\ast X^k_s\}_s$ is pro-trivial. Are there reasonable conditions under which we can conclude that the same is true after taking totalizations? That is, are there conditions under which $\{\pi_\ast \mathsf{Tot}X^\bullet_s\}_s$ is pro-trivial?
It seems like strong convergence of each homotopy spectral sequence should be enough, but I haven't been able to prove this. I'm wondering if anyone has any suggestions for suitable conditions.
