  Consider the moduli space, $\mathcal{M}_{3},$ of cubic polynomials over
$\mathbb{C}$, with a marked critical point. Let $\mathscr{S}_{k,n}$ 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 $\mathscr{S}_{k,n}$ are irreducible, for all $k \geq 0, n>0$. In this
article, we show the irreducibility of eventually $2$-periodic curves, i.e.
$\mathscr{S}_{k,2},\; k\geq 0$ curves. We also note that the curves,
$\mathscr{S}_{k,2},\; k\geq 0$, exhibit a possible splitting-merging phenomenon
that has not been observed in earlier studies of $\mathscr{S}_{k,n}$ curves.
Finally, using the irreducibility of $\mathscr{S}_{k,2}$ curves, we give a new
and short proof of Galois conjugacy of unicritical points lying on
$\mathscr{S}_{k,2}$, for even natural number $k$.
