If (X,T) is a minimal system uniquely ergodic with $\mu$. Is there $p\in X$ such that $\mu(\partial B(p,t))=0$ for all $t>0$ for some metric d (with same topology)?.

This question is motivated by the answer of Anthony Quas in Balls in minimal systems

If $(X,T)$ is a minimal system uniquely ergodic with $\mu$. Is there $p\in X$ such that $\mu(\partial B(p,t))=0$ for all $t>0$ for some metric $d$ (with the same topology)?

This question is motivated by the answer of Anthony Quas in Balls in minimal systems.