Let $(P,\leq)$ be a partially ordered set (poset). We define the *ordering dimension* $\textrm{dim}_\textrm{ord}(P)$ of $(P,\leq)$ to be the smallest cardinal $\kappa$ such that there exist a set of linear orders $\mathcal{L}$ on $P$ with $|\mathcal{L}|=\kappa$ such that the ordering relation $\leq$ equals $\bigcap \mathcal{L}$. (It follows from Szpilrajn's theorem that the intersection of the collection of all linear orders extending $\leq$ equals $\leq$, so $\kappa$ is well-defined.)

Now the poset $(P,\leq)$ can be endowed with the interval topology $\tau_i(P)$, which is generated by the subbasis $$\{P\setminus \downarrow x: x\in P\} \cup \{P\setminus \uparrow x: x\in P\},$$ where $\downarrow x := \{p\in P: p\leq x\}$ and similarly for $\uparrow x$.

We denote the Lebesgue covering dimension $(P,\tau_i(P))$ by $\textrm{dim}_L(P)$.

Is there a poset $(P,\leq)$ such that $\textrm{dim}_\textrm{ord}(P) < \textrm{dim}_L(P)$?