The signed four-cycle detection conjecture for $L_q$ random geometric graphs

Let $n,d$ be positive parameters, let $q\in[1,\infty]$, and consider testing

$$H_0:\mathsf{G}(n,1/2)$$

against

$$H_1:\mathsf{RGG}(n,\mathbb{T}^{d},\mathsf{Unif},\sigma^q_{1/2},1/2).$$

Here the signed four-cycle test is the signed subgraph-count test based on the four-cycle $C_4$. Signed four-cycle detection conjecture. The signed four-cycle test distinguishes the two hypotheses with high probability in either of the following regimes:

1. $q=o(d/\log d)$ and $dq=\omega(n^2)$;
2. $q=\Omega(d/\log d)$ and $d=\tilde{\omega}(n)$.

The conjecture extrapolates the expected orders of the signed four-cycle count from the endpoint cases $q=2$ and $q=\infty$. Its status is not resolved in the supplied source.