There are mixed recommendations regarding carrying your passport in Uzbekistan:
According to UZtrips, "it is advised to carry your passport with you while you are in Uzbekistan, especially during independent sightseeing, transfers between cities, and at train and bus stations". They note that police have the authority to check your passport, which is a regular occurrence.
However, several travelers report different experiences. On Tripadvisor, one traveler mentions: "It is most unlikely that anyone will ask you for your passport. Much safer for your passport to be kept in the safe at your hotel". Another traveler confirms: "I just got back from Uzbekistan, never was it asked from us except upon hotel check-in."
The consensus among experienced travelers seems to be: