The ruling on one who enters the mosque before Fajr and performs Witr with one rak'ah: does it suffice for the greeting of the mosque?

In the hadith of the Prophet (peace and blessings be upon him): (When one of you enters the mosque, let him pray two rak'ahs) Sometimes this occurs in the Haram; a person enters late, before Fajr, and wants to perform Witr with a single rak'ah. Does the single rak'ah suffice, and is the implication of the hadith regarding the two rak'ahs that the lesser amount is accepted, such that one rak'ah suffices for the greeting of the mosque?
Yes.
Witr suffices in place of the two rak'ahs. The Prophet (peace and blessings be upon him) said "two rak'ahs" because that is the common case. When does a person enter specifically to perform Witr with one rak'ah? The rare case, as the scholars say, follows the common case.