OLYMPIC-REVENGE 2019 Problem 2

Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.