Bowen's entropy-conjugacy conjecture is true up to finite index

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For a topological dynamical system consisting of a continuous map f, and a (not necessarily compact) subset Z of X, Bowen (1973) defined a dimension-like version of entropy, hₓ(f,Z). In the same work, he introduced a notion of entropy-conjugacy for pairs of invertible compact systems: the systems (X,f) and (Y,g) are entropy-conjugate if there exist invariant Borel subsets X' of X and Y' of Y such that hₓ(f,X X') < hₓ(f,X), hY(g,Y Y') < hY(g,Y), and (X',f|ₓ') is topologically conjugate to (Y',g|Y'). Bowen conjectured that two mixing shifts of finite type are entropy-conjugate if they have the same entropy. We prove that two mixing shifts of finite type with equal entropy and left ideal class are entropy-conjugate. Consequently, in every entropy class Bowen's conjecture is true up to finite index.