The symmetric-point maximum conjecture for random geometric graph entropy

Let $H_t(\mathcal{G})$ denote the entropy of the random geometric graph distribution at parameter $t$, and let $\bar{p}_{\max}$ be the corresponding edge-probability parameter at which the entropy is maximized. The dimension is denoted by $d$. Symmetric-point maximum conjecture. The maximum of $H_t(\mathcal{G})$ occurs when $t=0$, and therefore

$$\bar{p}_{\max} \rightarrow \frac{1}{2}$$

as $d\rightarrow\infty$. Numerical simulations suggest that the stationary point at $\bar{p}=\frac{1}{2}$ is a global maximum, but the claim is not proved in the source; it concerns the asymptotic location of the maximum-entropy configuration in high dimensions.