Finite-sum conjecture for morphisms between sheets

Let $S$ and $S'$ be sheets, regarded as objects of $\operatorname{Rep}(\Lambda_{\mathbb{I}})$. Elementary morphisms are those constructed in Proposition~, and multi-elementary morphisms are sums of elementary morphisms with pairwise disjoint supports as defined in the paper. Finite-sum conjecture. Any morphism

$$\phi:S\to S'$$

in $\operatorname{Rep}(\Lambda_{\mathbb{I}})$ is a finite sum of multi-elementary morphisms. The conjecture would imply that the elementary and multi-elementary constructions account for all morphisms between sheets; the supplied text gives no resolution.