The birational simplex-like subdivision conjecture for positive geometries

Let $(X,X_{\geq 0})$ be a positive geometry. A subdivision is a collection of positive geometries $(X_1,X_{1,\geq 0}),\ldots,(X_r,X_{r,\geq 0})$ with pairwise disjoint interiors and union of their nonnegative parts equal to $X_{\geq 0}$. Birational simplex-like subdivision conjecture. Every positive geometry has a subdivision into positive geometries that are birationally simplex-like. Such subdivisions would extend the canonical-form triangulation framework beyond ordinary polytopes, but the source gives no general proof.