Smallness equivalence conjecture for differential associative pseudoalgebras

Let $A$ be an $X^{cop}$-algebra, and let $\operatorname{Diff} A$ be its associated unital differential associative pseudoalgebra. Call $A$ small if $\dim\operatorname{F}^n A<\infty$ for every $n$, and call $\operatorname{Diff} A$ small if all its unital current subalgebras are finite as $H$-modules.

Smallness equivalence conjecture. A unital pseudoalgebra $\operatorname{Diff} A$ is small if and only if $A$ is a small $X^{cop}$-simple algebra.

This conjecture is intended to translate finiteness of the filtration components of $A$ into a finiteness condition for the associated pseudoalgebra. The supplied text gives no resolution or further evidence.