The signed triangle 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

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

Assume $dq^3=\omega(n^3)$. The signed triangle test is the signed subgraph-count test based on the triangle $C_3$. Signed triangle detection conjecture. The signed triangle test has the following behavior:

1. If $q=o(d/\log d)$, it succeeds with high probability when $dq^3=o(n^3)$ and fails with high probability when $dq^3=\omega(n^3)$.
2. If $q=\Omega(d/\log d)$, it succeeds with high probability when $d=\tilde{o}(n^{3/4})$ and fails with high probability when $d^3=\tilde{\omega}(n^{3/4})$.

This conjecture extrapolates the signed-triangle behavior known at the endpoint regimes $q=2$ and $q=\infty$; the supplied source does not resolve the asserted transition behavior.