Chebyshev classification of positive fixed-point proportion

Let $f\in \mathbb{C}[x]$ be a complex polynomial, and let $G_\infty(\mathbb{C},f,t)$ denote its geometric iterated Galois group over a transcendental parameter $t$. A polynomial is linearly conjugate to another if it is obtained from it by conjugation by a linear polynomial. Chebyshev classification conjecture. The following are equivalent: $f$ is linearly conjugate to $\pm T_d$ for some $d\geq 2$, and

$$\mathrm{FPP}(G_\infty(\mathbb{C},f,t))>0.$$

Chebyshev polynomials are known to have positive fixed-point proportion, with value $1/2$ for odd degree and $1/4$ for even degree. The conjecture asserts that these, up to linear conjugacy and sign, are the only complex polynomials with positive fixed-point proportion.