My professor has submitted a letter of recommendation for me for graduate program admissions, and has listed that I did X and Y (specific activities) under him, when I was only involved in doing X. He shared the letter with me after he submitted it, that's how I know.
I emailed him saying that I only did X and gave him info on how to submit a new letter, and his response was that if the admission committee ever calls him to check, he will clarify.
I really worry about this because 1) committees seldom check, 2) it's a falsehood that I did Y under him, 3) I think he is forgetful, and may just not remember to correct himself if they ever check, and 4) if I don't convince him to change this it would be like I am complicit in the falsehood.
He's a chill, old dude, my professor, and I don't want to rub him the wrong way by saying "We would be lying, please submit a new letter".
What's a gentle, tactful way to convince him to submit a new letter of recommendation with the correct information? Or is it okay if he clarifies if they ever call him to check?