The MA$_{\aleph_1}$ non-implication for uncountable strongly surjective real suborders

Let $\textmd{MA}_{\aleph_1}$ denote Martin's axiom for partially ordered sets satisfying the countable chain condition at cardinality $\aleph_1$. An uncountable strongly surjective real suborder is an uncountable suborder of $\mathbb{R}$ such that every suborder is the order-preserving image of the whole order. The MA$_{\aleph_1}$ non-implication. $\textmd{MA}_{\aleph_1}$ does not imply the existence of an uncountable, strongly surjective real suborder. This is posed as a natural open problem concerning the interaction between Martin's axiom and strongly surjective suborders of the real line; the supplied text gives no resolution.