Irreducibility of eventually $2$-periodic curves in the moduli space of cubic polynomials

Consider the moduli space, ${\mathcal{M}}_3,$ of cubic polynomials over $\mathbb{C}$, with a marked critical point. Let ${\mathcal{S}}_{k,n}$ be the set of all points in ${\mathcal{M}}_3$ for which the marked critical point is strictly $(k,n)$-preperiodic. Milnor conjectured that the affine algebraic curves ${\mathcal{S}}_{k,n}$ are irreducible, for all $k\ge 0,n>0$. In this article, we show the irreducibility of eventually $2$-periodic curves, i.e. ${\mathcal{S}}_{k,2},k\ge 0$ curves. We also note that the curves, ${\mathcal{S}}_{k,2},k\ge 0$, exhibit a possible splitting-merging phenomenon that has not been observed in earlier studies of ${\mathcal{S}}_{k,n}$ curves. Finally, using the irreducibility of ${\mathcal{S}}_{k,2}$ curves, we give a new and short proof of Galois conjugacy of unicritical points lying on ${\mathcal{S}}_{k,2}$, for even natural number $k$.