Milnor's irreducibility conjecture for preperiodic curves of cubic polynomials

Let $f_{a,b}(z)=z^{3}-3a^{2}z+2a^{3}+b$ be a monic, reduced cubic polynomial with marked critical point $a$, and let $\a7\Sigma_{k,n}$ be the Zariski closure in the moduli space $\a7\mathcal{M}_{3}$ of the points $(a,b)$ for which $a$ is strictly $(k,n)$-preperiodic under $f_{a,b}$, where $k\geq 0$ and $n>0$. Milnor's conjecture. For every choice of integers $k\geq 0$ and $n>0$, the curve $\a7\Sigma_{k,n}$ is irreducible. The conjecture generalizes Milnor's assertion for the curves $\a7\Sigma_{0,n}$; the cases $n=1$ were proved by Buff, Epstein, and Koch, and this paper proves the cases $n=2$, while the general case remains open.