The product dimension conjecture for posets

Let $P$ and $Q$ be posets. Write $P\times Q$ for their Cartesian product, ordered componentwise, and let $\dim(P)$ denote the least integer $n$ such that $P$ is isomorphic to a subposet of $(\mathbb{R}^n,\leq)$. The product dimension conjecture.

$$\dim(P\times Q)\geq \dim(P)+\dim(Q)-2.$$

This is described as a long-standing conjecture in order-dimension theory; the supplied text gives no resolution, so its status is open.