Higher-genus non-orientable self-shrinkers with line boundary

Determine whether embedded higher-genus non-orientable self-shrinkers in \mathbb{R}^3 with boundary a line exist.

Background

The paper relates minimal surfaces in the three-sphere to self-shrinkers for mean curvature flow and cites an embedded self-shrinking Möbius band with boundary a line. It leaves open whether analogous self-shrinkers with higher non-orientable genus can be constructed.