Is there any "Brunnian-like" braid which is not a pure braid in $B_n$ with $n\geq 3$?

Question

A braid $\beta$ is called Brunnian if it satisfies

1. $\beta$ is a pure braid;
2. $\beta$ becomes a trivial braid after removing any of its strands.

Obviously in $B_2$, every braid satisfies condition 2. When $n\geq 3$, is there any braid that satisfy Condition 2 but not Condition 1?