Computable copies of compact left-c.e. Polish spaces

Determine whether every effectively compact left-c.e. Polish space has a computable, equivalently right-c.e., Polish copy.

Background

Known examples show that some left-c.e. Polish spaces have neither computable nor right-c.e. presentations, while left-c.e. Stone spaces do have computable Polish copies. Compactness is proposed as a possible sufficient condition.