The gold partition conjecture and the Lexicographic sum of posets

We prove that if a finite poset $Q$ satisfies the Gold Partition Conjecture, and $P$ is a finite poset, then for any $i\in P$ the lexicographic sum $P\circ_iQ$ of $P$ with $Q$ on the point $i$, satisfies the Gold Partition Conjecture. Let $L(P)$ be the set of linearizations of $P$, $e(P)=\#L(P)$,