2 problems

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

External-action simplicity conjecture. Whenever the action of $X^{cop}$ is external in this sense, $A$ is $X^{cop}$-simple if and only if $\operatorname{F}^0 A$ is simple.