Finite invariant-set conjecture for exactly solvable operators

Let $T: \mathbb{C}_n[x] \to \mathbb{C}_n[x]$ be an invertible linear operator and let $S \in {{\mathcal I}_{n}^T}$ contain at least two points and have finite cardinality. A finite invariant-set conjecture asserts that

$$T[f] = (cx+d)^n f\left(\frac{ax+b}{cx+d}\right),$$

where $\phi(x) \coloneqq (ax+b)/(cx+d)$ is a Möbius map satisfying $\phi^{k}(x)=x$ for some $k\geq 1$. In the special case where $T$ sends every degree-$n$ polynomial to a degree-$n$ polynomial, $T$ is induced by a rotation through a rational angle: after a linear change of variables,

$$T\left[\prod_{i=1}^n(z-a_i)\right]=\prod_{i=1}^n(\alpha z-a_i)$$

for some root of unity $\alpha$. This conjecture seeks to classify invertible operators admitting finite invariant sets with at least two points; the stated finite-cardinality hypothesis and the distinction between the general Möbius form and the degree-preserving special case are essential to the proposed classification.