Polytopal invariance for Lusztig's $q$-weight multiplicities

Let $Φ$ be an irreducible root system normalized so that long roots have length $\sqrt{2}$, and let $μ\leqλ$ be dominant weights. We associate to $(Φ,μ,λ)$ the convex polytope $$P^Φ_{μ,λ} = (μ+C)\cap(λ-C)\cap(\overline{D}-ρ),$$ where $C=\mathbb{R}_{\geq 0}Φ^+$ and $\overline{D}$ is the closed domina