Let $B_{\infty}$ denote the infinite strand braid group. Let $\text{sh}:B_{\infty}\rightarrow B_{\infty}$ be the homomorphism defined by $\text{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Give $B_{\infty}$ the self-distributive operation $*$ defined by $$x*y=x\cdot\text{sh}(y)\cdot\sigma_{1}\cdot\text{sh}(x^{-1}).$$

Does there exist a linear ordering $L$ along with a function $\Gamma:B_{\infty}\rightarrow L$ where

$\Gamma(x*y)\leq\Gamma(y)$ whenever $\Gamma(y)<\Gamma(x)$, and

$\Gamma(x*y)>\Gamma(y)$ whenever $\Gamma(y)\geq\Gamma(x)$.

The motivation behind this question is that if there is no such map $\Gamma$, then $B_{\infty}$ cannot be embedded into an algebra $\mathcal{E}_{\lambda}$ of rank-into-rank embeddings since the critical points provide such a mapping $\Gamma$. However, if there is such a mapping $\Gamma$, then it would be reasonable to conjecture that $B_{\infty}$ does embed into some $\mathcal{E}_{\lambda}$.