Lorenz satellite conjecture

A Lorenz knot is a knot arising as the closure of a Lorenz braid. A satellite knot is obtained from a pattern knot in a solid torus by embedding that solid torus as a tubular neighbourhood of a companion knot in $S^3$; the resulting knot is the satellite. When the pattern is a torus knot on a torus parallel to the boundary of the solid torus, the satellite is a cable knot.

Lorenz satellite conjecture. A Lorenz knot that is a satellite has companion a Lorenz knot. Its pattern, when embedded in an unknotted solid torus in $S^3$, is equivalent to a Lorenz knot in $S^3$.

The conjecture refines Morton's conjecture after the latter was disproved in its claim that satellite Lorenz knots must be cables on Lorenz knots. The companion assertion remains open, as does the stronger assertion that every satellite Lorenz knot has a Lorenz pattern.