The ordinal Ramsey bound conjecture for well-founded trees

A well-founded tree is a prefix-closed set of finite sequences, and its order type is the ordinal order type used in the paper. A coloring of such a tree assigns each node one of two colors, red or black. Ordinal Ramsey bound conjecture. There is a reasonably simple ordinal-valued binary partial function $F$ such that, for all ordinals $\beta$ and $\gamma$, some ordinal $\alpha\leq F(\beta,\gamma)$ has the property that every red-and-black coloring of every well-founded tree $T_0$ of order type $\alpha$ contains either a well-founded tree $T_1$ of order type $\beta$ homeomorphically embedded into the red nodes of $T_0$, or a well-founded tree $T_2$ of order type $\gamma$ homeomorphically embedded into the black nodes of $T_0$. This is posed to generalize the finite-order-type Ramsey bounds established earlier in the appendix; no resolution is given.