Here is a somewhat more explicit version of a question that I asked a while ago.

Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of $p-1$ be a residue of degree $2(n+1)$ modulo $p$? That is, can one have $q^n\equiv 1\pmod p$ for every odd prime $q\mid p-1$?

Heuristically, this is extremely unlikely, and a computer search yields no such primes in the range $1<n\le10^7$ (the case $n=1$ being a trivial exception). However, I got stuck trying to prove anything rigorously.

My guess is that this may be related to higher reciprocity laws, but at least the Eisenstein reciprocity law does not seem to help.