Largest family without a pair of posets on consecutive levels of the Boolean lattice

Suppose $k ge 2$ is an integer. Let $Y_k$ be the poset with elements $x_1, x_2, y_1, y_2, ldots, y_{k-1}$ such that $y_1 < y_2 < cdots < y_{k-1} < x_1, x_2$ and let $Y_k$ be the same poset but all relations reversed. We say that a family of subsets of $[n]$ contains a copy of $Y_k$ on consecutive levels if it contains $k+1$ subsets $F_1, F_2, G_1, G_2, ldots, G_{k-1}$ such that $G_1subset G_2 subset cdots subset G_{k-1} subset F_1, F_2$ and $|F_1| = |F_2| = |G_{k-1}|+1 =|G_{k-2}|+ 2= cdots = |G_{1}|+k-1$. If both $Y_k$ and $Y_k$ on consecutive levels are forbidden, the size of the largest such family is denoted by $mathrm{La}_{mathrm{c}}(n, Y_k, Y_k)$. In this paper, we will determine the exact value of $mathrm{La}_{mathrm{c}}(n, Y_k, Y_k)$.