Algebraic-or-real-closed conjecture for superrosy NIP fields

Let $T$ be a superrosy theory with NIP, and let $K$ be the field under consideration.

Algebraic-or-real-closed field conjecture. The field $K$ is either algebraically closed or real closed.

The conjecture is presented as a harder statement following the theorem that a superrosy field with fsg is algebraically closed. The source compares it with the conjecture that every supersimple field is pseudo-algebraically closed, but gives no resolution.