Positive Casimir energy conjecture for the (2,3,7)-triangle group orbifold

Let $\Delta(2,3,7)$ be the $(2,3,7)$-triangle group acting on the hyperbolic plane $\mathbb{H}$, and let $\ell_n$ denote the lengths of the relevant closed geodesics, subject to the assumption

$$\text{the equality specified by }$$

for those lengths. Positive Casimir energy conjecture. Under this assumption, the Casimir energy of the orbifold $\Delta(2,3,7) \setminus \mathbb{H}$ is greater than or equal to $0.01$. The claim would establish positive Casimir energy, and hence a repulsive Casimir force, for this hyperbolic orbifold with conical singularities; the supplied text does not state whether the claim has been proved or disproved.