Given a poset $(P,\leq)$ the *interval topology* $\tau_{\text{int}}(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.

Let $P, Q$ be posets and $e:P\to Q$ be order-preserving and surjective. Assume that $(P,\tau_{\text{int}}(P))$ is Hausdorff. Does $(Q,\tau_{\text{int}}(Q))$ have to be Hausdorff?