KöMaL 2019 B5068

Let $p$ be an at most 1998th-degree polynomial such that the values $p(1), p(2),\dots,p(2000)$ form a permutation of the numbers $1,2,\dots,2000$. Does it follow that $p(1)$ and $p(2000)$ are $1$ and $2000$ in some order?