Path-connected interval topologies

Question

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?

Answer 1

No. If $P$ is any set and $<$ is the empty relation then the interval topology is just the cofinite topology. Now if $P$ has size $\mathfrak c$ then $P$ with the cofinite topology is path-connected just because any biyection $[0,1] \to P$ is continuous.