Shelah's consistency conjecture for Specker orders and Aronszajn trees

Let a Specker order and the orders $I,J$ be as described immediately before the statement. Let $T$ be an Aronszajn tree, let $c:T\to 2$, and write $\leq_T$ for its tree order. Shelah's consistency conjecture. It is consistent that any Specker order contains a suborder as above, and that if $I,J$ are as above, then $I,J$ or $J^*$ have uncountable isomorphic suborders. Equivalently, for every Aronszajn tree $T$ and every coloring $c:T\to 2$, there are an uncountable $X\subseteq T$ and $\ell\notin\\{0,1\\}$ such that for every distinct $x,y\in X$, the maximal $z\leq_T x,y$ satisfies $c(z)=\ell$.

The source says that the second consistency assertion is proved by a cited construction; the supplied status evidence says the associated claim is resolved by constructing a very special Aronszajn tree.