Unitary vertex operator algebras and conformal nets are mathematical axiomatizations of roughly the same physical idea: a two-dimensional unitary chiral conformal field theory. From a mathematical perspective, the axiomatizations are quite different in nature, and physical ideas which have been rigorously proven in one framework sometimes remain quite difficult in the other. In this talk I will explain a positivity conjecture for unitary VOAs which arose in the course of ongoing work to compare the theory of tensor products (i.e. fusion of sectors) as it appears in both settings. The conjecture is not limited to rational VOAs, and it suggests a general construction of unitary tensor products of modules. As time allows, I will present ongoing work to relate the representation theory of conformal nets and VOAs, as well as work in progress to establish the positivity conjecture in a broad class of examples.