The signed-triangle threshold conjecture for spherical random geometric graphs

Let $\tau_{p,d}$ be chosen so that the spherical threshold connection ${\boldsymbol T}_p(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{1}[cobreak\langle\boldsymbol{x},\boldsymbol{y}\rangle\geq\tau_{p,d}]$ has edge probability $p$, and let $cobreak\mathsf{RGG}(n,\mathbb{S}^{d-1},\mathsf{Unif},p,\boldsymbol{T}_p)$ denote the corresponding random geometric graph. The signed-triangle threshold conjecture. The total variation distance satisfies

$$\mathsf{TV}\Big(\mathsf{G}(n,p),\mathsf{RGG}(n,\mathbb{S}^{d-1},\mathsf{Unif},p,\boldsymbol{T}_p)\Big)=o_n(1)$$

when $d=\widetilde{\Omega}(n^3p^3)$ for $p=\Omega(1/n)$. The signed-triangle statistic is believed to give the tight indistinguishability threshold, but the source does not provide a resolution of this conjecture.