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@user I donāt have the time to look for the right argument now but my intuition is this: If $\sum_k S^k$ were to go to $\infty$, then it seems āhard for $V^k$ to remain non-negative for all $k$.ā
Indeed as @Dieter Kadelka said, taking the trivial filtration works. But what you are trying to show should be provable also with much more elementary methods.
@Daniele This is a great suggestion! Thank you for making me aware of this. Maybe some time I will try to use this, but for now I've had enough of this problem :)
