It is known that minimality does not imply unique ergodicity (Furstenberg example). I ask if the implication holds in following particular situation:

Suppose $X$ is a compact space, $f:X \to X$ is a minimal and uniquely ergodic homeomorphism. Suppose $p: Y \to X$ is a 2-to-1 covering map, and that $g:Y \to Y$ is a homeomorphism covering $f$, i.e., $p \circ g = f \circ p$. Suppose $g$ is minimal. Does it follow that $g$ is uniquely ergodic?

Example: If $f$ is an irrational rotation of the circle then it is easy to see that so is $g$, and the answer is yes in this case.