Forbidden induced subposets in the grid

In this short paper, we prove the following generalization of a result of Methuku and P\'{a}lv\"{o}lgyi. Let $P$ be a poset, then there exists a constant $C_{P}$ with the following property. Let $k$ and $n$ be arbitrary positive integers such that $n$ is at least the dimension of $P$, and let $w$ be the size of the largest antichain of the grid $[k]^{n}$ endowed with the usual pointwise ordering. If $S$ is a subset of $[k]^{n}$ not containing an induced copy of $P$, then $|S|\leq C_{P}w$.