Liquid input prioritization not working

I have two water sources supplying my electrolyzers. One comes from a pool, which I would prefer not to exhaust until I have to. The other comes from my water sieve, which I would like to use as much of as possible to avoid generating excess polluted water. I have seen in the past that a way to do this is to have a bridge from the high priority source with the output running over the pipe with the low priority source. However, it seems that no matter how I arrange the bridges, all of the water is coming from the pool, which I don't want. See the screenshots below. Am I doing something wrong?