Conjecture on the maximal number of bi-infinite branches with a common past direction

Let $d$ be the dimension, let $\text{BI}$ denote the set of bi-infinite branches, and consider their asymptotic directions toward the past, represented by limits $\lim_{t\to -\infty}f(t)\in\mathbb{R}^d$. Maximal-branch conjecture. Almost surely, the maximal number of bi-infinite branches sharing the same asymptotic direction toward the past is $d+1$; that is,

$$\max_{x\in\mathbb{R}^d}\#\left\{f\in\text{BI}:\lim_{t\to -\infty}f(t)=x\right\}=d+1.$$

The question of how many bi-infinite branches can share a common past asymptotic direction is explicitly stated to be unsolved. The conjecture predicts an almost-sure sharp upper bound, attained by the maximal number of such branches.