Positivity conjecture for toroidal Casimir energy in unitary bosonic CFTs

Let $d\geq 3$ and let the theory be a unitary bosonic conformal field theory that is not a topological quantum field theory. Let $\mathcal{E}(\tau)$ denote its dimensionless modular-invariant toroidal Casimir energy density, as defined by the relation between the vacuum energy density and $\mathcal{E}(\tau)$. Positivity conjecture. The function $\mathcal{E}(\tau)$ is strictly positive for every point $\tau$ in the upper half-plane. This is motivated by the positivity observed in all examples studied in the paper, although a general proof is not given.