homological algebra - Colimit of intersections

Let $B_i^p$ be a family of sets, where $pin mathbb{N}$ and $i in I$, $I$ being a directed set, and such that, for every $i$, we have a descending chain of inclusions

$$dots supset B_i^{p-1} supset B_i^p supset B_i^{p+1} supset dots$$

Question: is the following implication true?

$$bigcap_p B_i^p = emptyset, text{for all} i quad Longrightarrow quad bigcap_p varinjlim_i B_i^p = emptyset .$$

Since $bigcap_{}$ is a limit, this seems a problem of an interchange of limits and filtered colimits and indeed there is a universal map

$$varphi: varinjlim_i bigcap_p B_i^p longrightarrow bigcap_p varinjlim_i B_i^p$$

If $varphi$ was a bijection, then my implication would be true with no doubts,
but, since the intersection is not finite, I can not say that $varphi$ is a bijection. Nevertheless, could my implication still be true, without $varphi$ being a bijection?

The reason behind my question is the following: let $(A_i, F_i)$ be a directed family of filtered sets (or abelian groups, or modules; in fact, in my problem they are cochain complexes). Since filtered colimits (direct limits) are exact, you can define a filtration on the colimit like this:

$$F^pvarinjlim_i A_i = varinjlim_iF_i^pA_i .$$

Now assume all the filtrations $F_i$ are Hausdorff; that is, $bigcap_p F_i^pA_i = 0$ for all $i$. Is it then necessarily true that the filtration $F$ on $varinjlim_iA_i$ is Hausdorff too?

This question is a sequel to my previous question Convergence of right half-plane spectral sequence bounded on the right . Despite Tilman's counterexemple to my guess there, I think I've managed almost to prove it because my spectral sequences are right half-plane and this is the final detail I need.

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