Conjecture about prime numbers

Question

$$\forall k\in\mathbb{N},k\ge1,\exists p:k^3\lt p\lt (k+1)^3$$ with $p$ prime number. In other words is it possible to prove that for every $k\gt1$, with $k$ integer number it exists a prime number between $k^3$ and $(k+1)^3$? Legendre conjecture states there is a prime number between $k^2$ and $(k+1)^2$, but it's unsolved. Thanks.

Answer 1

Yes, it has been recently shown that there is at least one prime between every two consecutive cubes $x^{3}$ and $(x+1)^{3}$ if $\log (\log (x))≥15$, see the paper of Cheng of $2013$. Dudek has shown the result for all $x\ge \exp(\exp(33.217)$ and discusses some issues in the paper of Cheng. For the paper of Dudek see here.

Edit: As pointed out in the comments, this solves the question for almost all $x$, but not for all.