@Paul A. - Written communication on the internet is dangerous because you can't see tone of voice, eye contact, etc. It is sometimes difficult to tell what someone means.
When Jeff in Tx gave his original answer, it was difficult to tell, not being able to see or hear him, whether he meant that 1) Lexus LS never have problems, so I must not have a mildew problem, or 2) Lexus LS may have mildew problems, but the dealer's remedy made no sense. After corresponding with him, it is clear that he meant that the Lexus Dealer's solution made no sense. He noted that this happens on multiple manufacturers, and he shared an inexpensive remedy. I am grateful for his help on two levels. 1) If the dealer fails to fix the problem, I have an inexpensive solution to try. I am hoping that by the end of the 4 year warranty they will figure it out. and 2) I have a strong sense that fussing with the recirculate button has nothing to do with the problem, so I don't need to worry about it so much.
Anyhow, thanks for your help Jeff.