Gauging the strength of Fraisse's order-type conjecture

We seek to find the exact logical strength of a certain mathematical theorem commonly known as Fraisse's Conjecture, using the method of Reverse Mathematics. The usual pattern in mathematics is that, in order to establish that a theorem is true, we must obtain a proof. The proof will consist of a few self-evident statements called "axioms," from which we logically deduce new statements, and so on, until eventually we arrive at the theorem in question. This type of proof has been the gold standard in mathematics ever since the time of Euclid. But once the work is done, the efficiency-minded may begin to wonder if indeed we have made the SHORTEST proof, or the most UNDERSTANDABLE proof, or the one needing the FEWEST AXIOMS.
From our point of view, the simplest proof is the one with the fewest and simplest axioms, and the simplest theorems are the ones with the simplest proofs. When we say we want to gauge the strength of Fraisse's Conjecture, we mean that we want to find the simplest possible axioms that are able to prove it. The method of Reverse Mathematics is as follows: If we are able to use the axioms A to prove the theorem F, but also to reverse the roles and use F to prove the A, then A is exactly what is needed to prove F. This mode of thinking should be familiar to any mathematician who has seen the equivalence of the Axiom of Choice and Zorn's Lemma.
We have identified a few candidate axiom systems already.