The $SOP_2$ characterization of maximality in Keisler's order

Let $EC$ be an elementary class with the strict order property $SOP_2$, meaning that there are a model $M\in EC$, a first-order formula $\varphi(x;y)$, and parameters $\{a_\eta:\eta\in{}^{\omega>}2\}\subseteq{}^{\ell(y)}M$ such that, for every $1\leq n<\omega$ and $\eta_1,\ldots,\eta_n\in{}^{\omega>}2$, the set

$$\{\varphi(x;a_{\eta_i}):1\leq i\leq n\}$$

is consistent if and only if $\eta_1,\ldots,\eta_n$ lie along a single branch. The $SOP_2$ maximality conjecture. $SOP_2$ characterizes maximality in Keisler's order. The question concerns the model-theoretic characterization of the maximum class in Keisler's order, whose existence was established by Keisler. The preceding problem asks whether $SOP_2$ implies maximality; the conjecture asserts the corresponding characterization, and the text provides no resolution.