Context
I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and describes any progress on the problems. The book was published in 1995, so I don't know if some of the problems that Kirby stated are still open or proven. There is one problem that stood out for me.
Problem 1.8 (Stallings)
Suppose $\beta$ is a word in the generators $\sigma_1 , . . . , \sigma_{n−1}$ and their inverses in the braid group $B_n$. If the length of $\beta$ is minimal over all words representing the same element of $B_n$, call β minimal.
Conjecture: If the last letter of a minimal word $\beta$ is $\sigma_i^\epsilon$, then the word $\beta\sigma_i$ is again minimal $(\epsilon = \pm1)$.
Question My first question is a matter of understanding the statement. What does it mean to be minimal? Is it saying that for a word $\beta$ with a last letter of $\sigma_i^\epsilon$ and an inverse in the braid group, if it's in its reduced form representing the same element of $B_n$ (which doesn't make sense because how can a reduced word represent the same element of a group?) then $\sigma_i$ is again minimal? If $\epsilon=-1$, then how can $\beta\sigma_i$ represent the same element of the group as well? My misunderstanding is with the term minimal.
My second question is about the status of the conjecture. Kirby states that this is still an open problem. Has this conjecture been proven since then? I tried to look for stalling conjectures on the internet, but I couldn't find a webpage that talks about this problem. Is there a different name for this conjecture?