Characterization of posets embeddable in the nonnegative reals

Let $(Q,{\preccurlyeq}_{Q})$ be a partially ordered set. An extension to a total order is a total order ${\preccurlyeq}$ on $Q$ extending ${\preccurlyeq}_{Q}$. A totally ordered subposet is a subset of $Q$ equipped with the order induced by ${\preccurlyeq}_{Q}$.

Poset extension conjecture. The following conditions are equivalent:

1. $(Q,{\preccurlyeq}_{Q})$ admits an extension to a totally ordered set $(Q,{\preccurlyeq})$ such that $(Q,{\preccurlyeq})$ is order-isomorphic to a subposet of $(\mathbb{R}^{+},\leqslant)$.
2. $|Q|\leqslant 2^{\aleph_{0}}$ and every totally ordered subposet of $(Q,{\preccurlyeq}_{Q})$ can be embedded into $(\mathbb{R}^{+},\leqslant)$.

The conjecture asks for an order-theoretic characterization of those posets that have a real-valued linear extension. The cardinality bound is necessary for such an embedding, while the condition on chains captures the order-theoretic obstruction; whether these conditions are sufficient is the open problem posed in the paper.