Milnor's Galois-orbit conjecture for unicritical preperiodic parameters

For $d\geq 2$, consider the unicritical polynomials

$$f_c(X)=X^d+c,$$

where $c\in\mathbb{C}$ and $0$ is the unique finite critical point. For integers $k\geq 0$ and $n>0$, consider the values of $c$ for which $0$ is strictly $(k,n)$-preperiodic under $f_c$.

Milnor's Galois-orbit conjecture. These values of $c$ form one Galois orbit under the action of the absolute Galois group of $\mathbb{Q}$.

This conjecture concerns the arithmetic structure of unicritical parameter spaces. The paper proves it for $n=1$ and odd prime degree, using the irreducibility result for the corresponding moduli-space locus; the stated general case remains open.