Birman–Williams braid conjugacy conjecture for isotopic Lorenz orbits

Let $b$ and $b'$ be Birman–Williams braids associated with two isotopic periodic orbits of the Lorenz flow. If $t$ is the common period of these orbits, then the Birman–Williams braid conjugacy conjecture. $b$ and $b'$ are conjugate in $B_t$.

This conjecture would provide a first step toward recognizing when two Young diagrams encode the same Lorenz knot. It has been verified in a large number of examples, but its general status is left open here.