No odd prime power is super Carmichael

A super Carmichael number is a weak Carmichael number satisfying

$$\sum_{\substack{1\le k\le n-1\gcd(k,n)=1}}k^{n-1}\equiv\varphi(n)\pmod{n^2}.$$

No odd prime-power super Carmichael conjecture. For every odd prime $p$ and every $f\ge 2$, the prime power $p^f$ is not a super Carmichael number. The source gives a Bernoulli-number characterization for $p>3$, but no resolution of the conjecture.