The conjugacy problem for braid groups $B_3$ and $B_4$ can be solved in polynomial time, it is noted in the paper by Birman, Ko and Lee(2001).

That was a result in 2001. Are there any new results on other small braid groups? Is the conjugacy problem in $B_5$ solvable in polynomial time?

Also, I'm still curious on exactly how fast the conjugacy problem in $B_3$ can be solved(polynomial time is too broad). The only paper that described such algorithm is behind a paywall.