  We solve a long-standing conjecture by Barker, proving that the minimal and
maximal tensor products of two finite-dimensional proper cones coincide if and
only if one of the two cones is generated by a linearly independent set. Here,
given two proper cones $C_1$, $C_2$, their minimal tensor product is the cone
generated by products of the form $x_1 \otimes x_2$, where $x_1 \in C_1$ and
$x_2 \in C_2$, while their maximal tensor product is the set of tensors that
are positive under all product functionals $f_1 \otimes f_2$, where $f_1$ is
positive on $C_1$ and $f_2$ is positive on $C_2$. Our proof techniques involve
a mix of convex geometry, elementary algebraic topology, and computations
inspired by quantum information theory. Our motivation comes from the
foundations of physics: as an application, we show that any two non-classical
systems modelled by general probabilistic theories can be entangled.
