Infinite counterexamples to the finite-poset theorem

A locally finite poset is a poset in which every interval is finite. The preceding finite-poset theorem has several parts whose conclusions depend on finiteness. Infinite-counterexample conjecture. There exist infinite locally finite posets that are counterexamples to the parts of that theorem mentioned above. The source describes this as a belief and proposes seeds for counterexamples, so the claim remains open in the supplied text.