Triangular-torus minimum conjecture for toroidal Casimir energy

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 on the standard $PSL(2,\mathbb{Z})$ fundamental domain. Triangular-torus minimum conjecture. The function $\mathcal{E}(\tau)$ has a global minimum at $\tau=e^{2\pi i/3}$. The claim is supported by the free scalar, critical $O(N)$, and holographic examples discussed in the paper, but no general proof is supplied.