Existence of an asymmetric L-space knot with bridge index three

Determine whether there exists an asymmetric L-space knot with bridge index 3.

Background

The paper constructs infinitely many asymmetric hyperbolic L-space knots with every braid index, and hence bridge index, at least 5; previously known constructions realize bridge index 4. The authors note that the existence of an asymmetric L-space knot with bridge index 3 is unresolved. They further explain that this question is connected to a conjecture asserting equality of bridge index and braid index for all L-space knots.