Piecewise conjugacy conjecture for tensor algebras of multivariable systems

Let $(X,\f8\sigma)$ and $(Y,\f8\tau)$ be multivariable dynamical systems, and let ${\mathcal{A}}(X,\sigma)$ and ${\mathcal{A}}(Y,\tau)$ denote their tensor algebras. Piecewise conjugacy conjecture. The tensor algebras ${\mathcal{A}}(X,\sigma)$ and ${\mathcal{A}}(Y,\tau)$ are isomorphic if and only if the systems $(X,\sigma)$ and $(Y,\tau)$ are piecewise topologically conjugate; in this case, they are completely isometrically isomorphic. The theorem preceding this conjecture proves the equivalence under several hypotheses, including $n\leq 3$, covering dimension at most $1$, or absence of interior in the branch set; the general case remains open.