Legendre`s conjecture states that there is always a prime between $n^2$ and $(n+1)^2$ for every natural $n$.

It is natural to create following generalization:

Is it true that for every $\varepsilon \in (0,1]$ there exists natural number $n_0(\varepsilon)$ such that for every $n \geq n_0(\varepsilon)$ there is always a prime between $n^{1+\varepsilon}$ and $(n+1)^{1+\varepsilon}$?

I do not know if some results from number theory do not allow this generalization to be true but if it is not known is it true or not what is known, if anything, about this generalization of Legendre`s conjecture?