The exponential lower-bound conjecture for sphere percolation thresholds

Let $d\geq 3$ and let $\sigma_d$ denote the dimension-dependent constant used in the definition of the sphere-percolation model. Write $p_c^\mathcal{S}$ for its critical probability.

Exponential lower-bound conjecture. For every dimension $d\geq 3$,

$$p_c^\mathcal{S}\geq \frac{\exp(2\sigma_d)}{(2d-1)^2}>\frac{1}{8(d-1)}.$$

This conjecture would follow from the proposed infinitely fine partition argument if the corresponding value of $L$ were shown to satisfy $L<1$. It gives a closed-form lower bound for the critical probability, strengthening the general bound obtained earlier in the paper.