Criticality conjecture for multi-particle reinforced interacting random walks

Let $D$ be the state space of pairs of occupation measures, let $x(n),y(n)$ denote the two occupation-measure processes, let $z(n)=(x(n),y(n))$, and let $U$ denote the uniform occupation measure. For a given initial condition $(x(0),y(0))\in D$, the parameter $\alpha_c=1$ is critical. Criticality conjecture. If $\alpha>1$, there exists a constant $c=c(\alpha,d)$ such that, for any sufficiently small positive $\delta$,

$$\mathbb{P} \left\{ \exists N, \bigcap_{n\ge N}\left\{ \sum_{i=1}^d x_i(n)y_i(n)\le c\delta \right\} \right\}=1.$$

If $0<\alpha<1$, then

$$\mathbb{P} \left\{ \lim_{n\to \infty}z(n)=(U,U) \right\}=1.$$

The claim identifies $\alpha=1$ as the transition between the regime in which the two occupation measures asymptotically have small overlap and the regime in which both converge to the uniform measure. The supplied text does not establish the conjecture or provide evidence of its resolution.