Combinatorial-dimension characterization of universal Donsker classes

Let $F$ be a uniformly bounded class of functions, and let ${\it v}(F,t)$ denote its combinatorial dimension at scale $t\geq 0$: the maximal cardinality of a subset that is $t$-shattered by $F$. A class is universal Donsker when it is Donsker for every probability law, uniformly over all such laws.

Combinatorial-dimension conjecture. For every uniformly bounded class $F$,

$$\int_0^\infty \sqrt{{\it v}(F,t)}\,dt<\infty \quad\Rightarrow\quad F\text{ is universal Donsker} \quad\Rightarrow\quad {\it v}(F,t)=O(t^{-2}).$$

The conjecture proposes an optimal description of universal Donsker classes in terms of how much the class oscillates. The first implication gives a sufficient integrability condition, while the second gives a necessary polynomial bound on the combinatorial dimension; the source does not state that either implication is resolved in full generality.