The improved bound conjecture for primes between consecutive squares

For a positive integer $n$, let $leg(n)$ denote the number of primes between $n^{2}$ and $(n+1)^{2}$. The improved bound conjecture for $leg(n)$.

$$\frac{n^{2}+10n+5}{3n\ln(n)}\leq leg(n)\leq\frac{n^{2}+10n+5}{3n}.$$

The source proposes this as a tighter empirical bound than the previously derived upper bound for $leg(n)$; its validity remains open.