I would say no. The 'pollution' seems to be referring to some sort of enchantment or magic on the stream, and once out of the area of Minas Morgul, maybe the water would be safe again.
Perhaps a comparison could be made with the enchanted stream in Mirkwood. It flowed into the Forest River, which in turn went to the Long Lake and the River Running. The Lake-men apparently were not troubled by sudden urges to sleep.
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