The two-dimensional Poisson SINR percolation upper-bound conjecture

Let $X^\lambda$ be a homogeneous Poisson point process on $\mathbb{R}^2$, so that $\Lambda\equiv |\cdot|$, and let $g_{(\gamma,N_0,\tau)}(X^\lambda)$ be the SINR graph with noise parameter $N_0\geq 0$, threshold $\tau>0$, and intensity $\lambda>0$. Let $\gamma^*(\lambda)$ be its critical SINR parameter. The two-dimensional Poisson SINR percolation upper-bound conjecture. For every $N_0\geq 0$ and $\lambda>0$,

$$\gamma^*(\lambda)\leq \frac{1}{4\tau},$$

and

$$\mathbb{P}\left(g_{(\frac{1}{4\tau},N_0,\tau)}(X^\lambda)\text{ percolates}\right)=0.$$

The conjecture is derived from the proposed comparison with the bidirectional $k$-nearest-neighbour graph and the high-confidence claim that this graph percolates only for $k\geq 5$. The nearest-neighbour input is not a proof in the source because it relies on numerical evaluation of high-dimensional integrals; the resulting SINR statement is therefore open there.