Properness conjecture for the completed relative Ginzburg dg algebra

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\widehat{\mathbf{\Gamma}}^{[a,b]}(\underline{w}_0)$ be the completed relative Ginzburg dg algebra associated with the ice quiver with potential constructed from these data. Properness conjecture. The relative Ginzburg dg algebra $\widehat{\mathbf{\Gamma}}^{[a,b]}(\underline{w}_0)$ is proper. Properness is needed for the construction of the associated Higgs category and its additive categorification; the conjecture is posed here without evidence of a resolution.