Todorčević's Tukey basis conjecture for binary relations

Let $R$ be a binary relation. Write $\fulldiamond$ for the proper forcing axiom appearing in the source, and let $\beta\to(\beta_1;\beta_2)^2_2$ denote the corresponding partition relation. Todorčević's Tukey basis conjecture. In the presence of $\fulldiamond$, either

$$R\leq\aleph_0\cdot\omega_1$$

or

$$[\omega_1]^{<\aleph_0}\leq R.$$

This is a basis conjecture for binary relations on $\omega_1$ in the Tukey order; the paper presents it as a conjecture associated with Todorčević, and the supplied material gives no resolution.