Entangleability of cones

We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if only one is generated linearly independent set. Here, given $${\mathcal {C}}_1$$ , {C}}_2$$ their product cone form $$x_1\otimes x_2$$ where $$x_1\in {\mathcal $$x_2\in while set tensors are positive under all functionals $$\varphi _1\otimes \varphi _2$$ _1|_{{\mathcal {C}}_1}\geqslant 0$$ _2|_{{\mathcal {C}}_2}\geqslant . Our proof techniques involve mix convex geometry, elementary algebraic topology, computations inspired quantum information theory. motivation comes from foundations physics: as an application, we show any non-classical systems modelled general probabilistic theories can be entangled.