Domination decomposition conjecture for invariant types in o-minimal theories

Let $T$ be an o-minimal theory and let $\mathfrak U\models T$ be a monster model. A type over $U$ is invariant over a small set if it is invariant over some small subset of the monster model. A product of $1$-types is a Morley product of types in one variable. Domination decomposition conjecture. Every type over $U$ that is invariant over a small set is domination-equivalent to a product of $1$-types. The property is known for real closed fields, divisible ordered abelian groups, and o-minimal theories with no non-simple types in the stated sense; it is conjectured to hold for every o-minimal structure.