Smooth numbers in short intervals conjecture

For real $x,y$, let $\Psi(x,y)$ denote the number of positive integers at most $x$ all of whose prime factors are at most $y$. Such integers are called $y$-smooth numbers. Let $\epsilon>0$. Smooth numbers in short intervals conjecture. For sufficiently large $x$,

$$\Psi(x+x^\epsilon,x^\epsilon)-\Psi(x,x^\epsilon)\gg x^\epsilon.$$

This predicts that every sufficiently large interval of length $x^\epsilon$ contains many integers whose prime factors are at most $x^\epsilon$. The conjecture is stated as open in the source.