Finiteness conjecture for Type (i) and Type (iia) power mappings

Let $n$ be a positive integer, let $p$ be a prime with $p>2n$, and partition the reduced residues modulo $p$ into residue classes $I_i$ modulo $n$. Let $f(x)=Ax^k\pmod p$ be a power permutation. Type (i) means that $f(I_i)=I_i$ for every $i$, while Type (iia) means that $f(I_i)=I_{\sigma(i)}$ for some permutation $\sigma$ of $\{0,1,\ldots,n-1\}$. Finiteness conjecture. Excluding $f(x)=x\pmod p$, there are only finitely many Type (i) examples. Excluding $f(x)=\pm x\pmod p$, there are only finitely many Type (iia) examples. The conjecture formalizes the observed sparsity of these mappings beyond the trivial examples; the supplied text does not state a resolution.