The phrase "as believed etc" is commonly used to convey that something is believed to be true but may not necessarily be a proven fact. However, there are several synonyms that can be used to express a similar sentiment. These include "allegedly," which implies that something is claimed to be true but has not been confirmed. "Supposedly" suggests that something is believed to be true but there may be doubts or uncertainties surrounding it. "Reportedly" conveys that something is said to be true but the accuracy of the information is unconfirmed. Other alternatives include "presumably," "apparently," and "ostensibly," all of which indicate that something is perceived to be true but may not be entirely certain.