Asymptotic rarity conjecture for non-simple tile labels

Let $T$ be a tile of a weakly robust periodic solution (WRPS) with temporal period $\tau$ and spatial period $\sigma$. Set $x=\sigma/\gcd(\tau,\sigma)$, and let $\ell=p(T)-s(T)$. Assuming the lower-bound conjecture for tile ranks, let $I\subset\{0,\ldots,\sigma-1\}$ be the index set of size $x-\ell$ whose labels are the leftmost $x-\ell$ labels without a repeated state. Write the labels as $A_i$.

Asymptotic label conjecture. There exists an index $j\notin I$ such that

$$\mathbb{P}\left(A_j\Rightarrow A_{j+1}\mathrel{\bigm|}\{A_i\Rightarrow A_{i+1}\text{ for all }i\in I\}\right)=o(1).$$

The conjecture is intended to provide the additional asymptotic estimate needed to prove the proposed generalization of the probability asymptotics when $\sigma$ does not divide $\tau$. The supplied status evidence states that it remains open even in the cases discussed there.