Completely Li--Yorke chaotic homeomorphisms with positive entropy

It is an open problem whether a homeomorphism on a compact metric space satisfying that each proper pair is either positively or negatively Li--Yorke, called completely Li--Yorke chaotic, can have positive entropy. In the present paper, an affirmative answer to this question is given. In fact, for each homeomorphism $T$ with positive entropy such that each proper pair is not two-sided asymptotic, a completely Li--Yorke chaotic homeomorphism with positive entropy associated with the given homeomorphism can be constructed.