Beukers' conjecture for odd-prime equations $x^2-D=p^n$

Let $D$ be a fixed positive nonsquare integer, let $p$ be an odd prime, and let $N(-D,p)$ denote the number of solutions $(x,n)\in\mathbb{N}^2$ of $x^2-D=p^n$. Beukers' odd-prime conjecture.

$$N(-D,p)\leq 3.$$

This conjecture was completely proved by Bauer and Bennett, so it is solved.