Cover-graph incidence-poset doubling conjecture

Let $P$ be a poset, let $G$ be the cover graph of $P$, and let $Q$ be the incidence poset of $G$. Cover-graph incidence-poset doubling conjecture. For every $t\ge1$, there exists a poset $P$ such that

$$\dim(Q)=2\dim(P).$$

The text notes that this holds when $t\le2$ but reports that the issue for larger values of $t$ remains unsettled.