Boundedness or injectivity of non-escaping ends

Determine whether non-escaping ends of bi-rotational self-shrinkers are necessarily bounded, or whether non-escaping ends can be injective.

Background

The paper proves that every non-cylindrical end is either asymptotically conical or non-escaping, where non-escaping means infinite arclength without radial escape to infinity. The recurrence theorem shows that non-escaping behavior is prevalent in the measure-theoretic sense, but the geometric structure of such ends is not fully characterized. In particular, the authors do not resolve whether these ends remain in a bounded region or can be injective while repeatedly returning to the bounded core.