The barrier-cone characterization of bounded cluster-growth directions

For $p\in[0,1]$, let $\nu_p=p\delta_0+(1-p)\delta_1$ be the passage-time law, let $\mu_p$ be the associated semi-norm on $\mathbb{R}^d$, and define

$$A_p=\{x\in\mathbb{R}^d:\;\mu_p(x)\le 1\}.$$

The barrier cone of $A_p$ is

$$\operatorname{Bar}(A_p)=\{u\in\mathbb{R}^d:\;\sup_{x\in A_p}\langle x,u\rangle<+\infty\},$$

and, writing $C_+(0)$ for the oriented percolation cluster issued from $0$, define

$$\mathrm{BG}(p)=\left\{u\in\mathbb{R}^d:\;\mathbb{P}_p\left(\sup_{y\in C_+(0)}\langle y,u\rangle=+\infty\right)=0\right\}.$$

Barrier-cone characterization conjecture. For every $p\in[0,1]$,

$$\operatorname{Bar}(A_p)=\mathrm{BG}(p).$$

The set $\mathrm{BG}(p)$ consists of the directions in which the growth of the cluster issued from $0$ is bounded, while the barrier cone describes the directions in which the associated convex set has bounded support. The conjecture proposes that these probabilistic bounded-growth directions are exactly characterized by the semi-norm $\mu_p$.