It is known that two Markov subshifts with the same entropy are "almost isomorphic" (up to a subset of measure 0) if the entropy is a logarithm of an integer (see R. L. Adler, L. W. Goodwyn, and B.Weiss. Equivalence of topological markov shifts. Israel J. Math, 27(1):49--63, 1977). Is it true (known) if the entropy is not a logarithm of an integer?
Update Basically I want to know if the result of AGW is true without the integer assumption. I would appreciate an answer of the form "yes plus reference" or "no plus reference" or "it is still unknown".