Chemical engineer here. The difference in temperature between your cold and hot water supplies is what the problem is, and I would imagine this is not a problem at all in modern plumbing systems. Your cold water supply is usually about 50-60 deg F (10-15 C) while your hot water supply us usually set to 140 F (60 C). Solubility of some lead salts in water are given in this table with lead chloride being about 0.8 g/100mL at 10 C and 1.98 g/100mL at 60 C, so about 2.5x more soluble. The rate itself is a more complicated relationship, but it increases rapidly as well. Temperature has a big effect on these things.
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