Let $X$ be an irreducible shift of finite type and let $\pi$ be a factor code from $X$ to a sofic $Y$. Suppose $y$ is a right transitive point of $Y$ and $\pi(u)=y$ for some $u\in X$. Given $u_0=a$ and a block $B$ of $X$, is there a point $x\in\pi^{-1}(y)$ such that $x_0=a$ and $B$ occurs in $x_{[0,\infty)}$? (Note that the stronger statement is true when $\pi$ is a finite-to-one code: a point is transitive if and only if it's image under $\pi$ is transitive, so $B$ actually occurs infinitely often in the right of $x$.) Thanks to the helpers.