Let $(X,T)$ be a minimal subshift. Can it happen that an endomorphism $\varphi\colon (X,T) \to (X,T)$ is almost 1-to-1 but not 1-to-1? Can it happen that a factor $\pi\colon (X,T) \to (Y,T)$ between minimal subshifts is almost 1-to-1 but not 1-to-1? I know that, for example, Toeplitz subshifts are almost 1-to-1 extensions of a Cantor system (some odometer), but in the subshift case I couldn't find anything on internet.

PD. Here, a factor $\pi\colon(X,T)\to(Y,T)$ is almost 1-to-1 if $\exists y \in Y$ such that $\#\pi^{-1}(y) = 1$. From this, one can prove that $\#\pi^{-1}(y) = 1$ for all $y$ in a residual set.