Let $X$ be a compact metric space and $T$ a continuous map from $X$ to $X$. The system $(X,T)$ is called topologically weakly mixing if the product system $(X\times X,T\times T)$ is topologically transitive. My question is that: If $(X,T)$ is minimal and topologically weakly mixing, whether there exists an ergodic measure $\mu$ on $(X,T)$ such that $(X,T)$ is weakly mixing as a measure-preserving system, which means $(X\times X,T\times T,\mu\times \mu)$ is ergodic?
I guess there might be a counter-example but I do not have any such example in mind, nor can prove it. If this is not true, is it possible to add some proper condition to make this true but not trivially true?