The conjecture that unstable NIP theories interpret infinite linear orders

Let $T$ be a first-order theory. Say that $T$ is NIP if it does not have the independence property, and say that $T$ is unstable if it is not stable. An infinite linear order is an infinite structure equipped with a definable strict total order.

Unstable NIP conjecture. If $T$ is an unstable NIP theory, then $T$ interprets an infinite linear order.

This is a long-standing conjecture about NIP theories. The paper proves that any theory interpreting an infinite linear order is not hereditarily G-compact; thus the conjecture would imply that every unstable NIP theory fails hereditary G-compactness.