Galvin's conjecture on two-colour homogeneous copies of the rationals

Let $K$ be a positive natural number, and let $c:[\mathbb R]^2 \rightarrow K$ be a colouring of pairs of reals. Galvin's conjecture. There is a set of reals $Y$ homeomorphic to $\mathbb Q$ such that

$$|c“[Y]^2| \leq 2.$$

This was an open problem attributed to Galvin and asked whether Galvin's theorem for order-isomorphic copies of $\mathbb Q$ could be strengthened to homeomorphic copies. The conjecture is verified by Corollary B of the paper.