Honor of Entering the Bathroom First

Hello Rabbi. Is there a specific source that suggests not honoring a great and respected person by allowing them to enter the bathroom first? If such a concept exists, is it a prohibition? Is it permissible to honor them if desired? Is this related to the Gemara in Berachot where they did not honor someone to enter a doorway not suitable for a mezuzah, and therefore, the bathroom door is also not suitable for a mezuzah, so there is no point in honoring them?! Thank you very much.

Berachot 47.
Rabbi Yochanan said: We only honor at a doorway that has a mezuzah. If it has a mezuzah, yes; if it does not have a mezuzah, no? Then, by this logic, in a synagogue and a study hall, where there is no mezuzah, do we also not honor? Therefore, say: at a doorway suitable for a mezuzah.

Beit HaBechira to Meiri Berachot 46:
And likewise, we do not honor at the doorway of a bathhouse and a bathroom, except at a doorway suitable for a mezuzah:

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