Given a braid group $$ B_n \simeq \left\langle x_1,\ldots,x_{n-1} \middle| \begin{array}{l} x_ix_j = x_jx_i, \;\text{for } |i-j|>1 \\ x_ix_{i+1}x_i = x_{i+1}x_ix_{i+1} \end{array} \right\rangle $$ for $n \geq 5$

- What is the centralizer of $x_1$?

or, less general question,

- Given a word $w = w(x_2,\dots,x_{n-1}) \in B_n$ such that $[x_1, w]=1$, does it imply that $w = w(x_3,\dots,x_{n-1})$?