Let $p\ge 11$ be a prime number, $n \ge 5$ be an odd positive divisor of $p-1$ and $s \in \mathbb Z_p$ such that $ord_p(s) = n$.

Is it true that the geometric progression $\{s^k\}_{k \in \mathbb Z_n}$ intersects some of the classes $\overline{p-n}, \;\; \overline{p-n+1}, \;\; \dots, \;\; \overline{p-1} \pmod p$?

Remark: since $n$ is odd, the class $\overline{p-1}$ is actually never achieved, so I could have written until the class $\overline{p-2}$.

Thanks!