A thought-experiment about the importance of Buddhist traditions

Suppose a leading Buddhist figure -- say a well-known Tibetan monk -- were to suddenly start displaying cultural behavior appropriate to other religious faiths. For instance, suppose this monk were to start performing namaz like a Muslim five times a day, and frequently make the sign-of-the-cross gesture during conversations. Would this result in loss of rank, social standing and credibility in his own circles? Or would this cause others around him to start looking up to him? What's your opinion? Are Buddhist cultural habits and traditional ways of dress, behavior, speech etc. important for making spiritual progress?