Galvin's partition relation conjecture at omega one

For a coloring of unordered pairs of $\omega_1$ with two colors, write $\omega_1\rightarrow(\omega_1;\omega_1)^2_2$ for the stated polarized partition relation. Galvin's partition relation conjecture.

$$\omega_1\rightarrow(\omega_1;\omega_1)^2_2.$$

The source attributes this conjecture to Galvin, while noting that the attribution is Hajnal's; the supplied material gives no resolution.