Equivalence of boundedness, dimensionality, and surjectivity of the restriction map

Let $T$ be a stable theory. Recall that $T$ is bounded if

$$\left|\operatorname{\widetilde{Inv}}(\mathfrak U)\right|<\left|\mathfrak U\right|,$$

and that $T$ is dimensional if every nonrealised global type $p$ has a global type $q$ that does not fork over the empty set and satisfies $p\centernot\perp q$. Let $\mathfrak e$ be the restriction map introduced in the paper.

Equivalence conjecture. The following are equivalent: $T$ is bounded; $T$ is dimensional; and $\mathfrak e$ is surjective.

The surrounding discussion establishes this equivalence in the thin case, while the statement is presented here for stable theories in general. The conjectural content concerns whether the correspondence between boundedness, dimensionality, and surjectivity persists without the thinness assumption.