The question is in the title. Here "$P$ embeds into $Q$" means there is a function $f : P\to Q$ such that for all $p,p'\in P$, $p \le_P p'$ if and only if $f(p) \le_Q f(p')$. A well quasi order $W$ is a relation which is reflexive and transitive, such that for any infinite sequence $w_1,w_2,\ldots,w_n,\ldots$ taken from the universe of $W$, there are some $n < m$ such that $w_n \le_W w_m$.
My question is motivated by this other recent question. Laver proved that the countable linear orders form a wqo under embeddability. If this holds more generally for countable partial orders, then that would imply that the $\bar{\mathcal{O}}$ from the question above has no infinite descending chain or infinite antichain.