Lifting generic points

Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi $ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu $ on X and $\nu $ on Y, with $\mu $ ergodic. Let $y\in Y$ be quasi-generic for $\nu $. Then there exists a point $x\in X$ generic for $\mu $ such that the pair $(x,y)$ is quasi-generic for $\xi $. This is a generalization of a similar theorem by T. Kamae, in which $(X,T)$ and $(Y,S)$ are full shifts on finite alphabets.