$\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$

Question

Conjecture

For $n \ge 1 $ , $m \ge 1$

$\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$

where $\pi\left(n\right)$ is the prime counting function .

---

Does this conjecture have a name ? Do you know of anywhere in the literature where this issue is investigated ?

---

P.S.

I know that there are similar conjectures like Legendre's conjecture and Brocard's conjecture .