Properness conjecture for relative Ginzburg dg algebras

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\mathbf{\Gamma}^{[a,b]}(\underline{w}_0)$ and $\widehat{\mathbf{\Gamma}}^{[a,b]}(\underline{w}_0)$ be the non-completed and completed relative Ginzburg dg algebras associated with the corresponding ice quiver with potential. Properness conjecture. The relative Ginzburg dg algebras $\mathbf{\Gamma}^{[a,b]}(\underline{w}_0)$ and $\widehat{\mathbf{\Gamma}}^{[a,b]}(\underline{w}_0)$ are proper. Properness is a foundational finiteness property for the relative Ginzburg algebras used in the paper's additive categorification; the source does not state a resolution of this conjecture.