Oppermann's conjecture on primes in quadratic intervals

Let $n$ be an integer with $n>1$. Oppermann's conjecture. There is at least one prime number between $n(n-1)$ and $n^2$, and at least another prime number between $n^2$ and $n(n+1)$. This is a strengthening related to Legendre's conjecture and remains open according to the supplied material.