Chromatic uniqueness of elements of height $\leq3$ in lattices of complete multipartite graphs

The purpose of the paper is to prove the following theorem. Let integers $n,t$, and $h$ be such that $0<t<n$ and $h\leq3$. Then, any complete $t$-partite graph with nontrivial parts that has height $h$ in the lattice $NPL(n,t)$ is chromatically unique.