This is a follow-up from this question.
For a finitely presented group $G=⟨S|R⟩$, a positive element g∈G is an element of G that can be written as a finite product of elements of $S$ only. A positive expression of $g\in G$ is a word $w$ with elements of $S$ its alphabet and only contain elements of $S$ and $w$ reduce to $g$ after applying relations in $R$. A relation $R$ is a positive relation if it can be rewritten as $w_1=w_2$ with $w_1$, $w_2$ positive elements. A basic positive replacement of $G$ is to replace $w_1$ with $w_2$ or $w_2$ with $w_1$ for some positive relation $w_1=w_2$. A group presentation $<S|R>$ is positively admissible if all its relations are positive relations and no element of $S$ is the inverse of any element of $S$ (This can be simplified to $S\cap S^{-1}=\varnothing$). A group $G$ is positively admissible if there exists some positively admissible presentation $G=<S|R>$. For a positively admissible presentation $<S|R>$ such that any positive expression w for some positive element g can be deformed into another positive expression w′ of g only by finite applications of basic positive replacements, we call g a reachable element with respect to $<S|R>$. If there exists a positively admissible presentation such that $g$ is a reachable element with respect to $<S|R>$, we call g a reachable element. A finitely presented positively admissible group with all elements reachable with respect to a particular $<S|R>$ (which is independent of what the element is) is called a uniformly reachable group.
What kind of conditions can we add so that a finitely presented positively admissible group is uniformly reachable? To be more specific, are braid groups reachable?
Edited: Some definitions have been changed since reachability is dependent on the presentation.