Sketchability of the planar lamplighter zero section

Determine whether the zero section \((_2\wr^2)_0\) of the planar lamplighter group is sketchable in the algorithmic sense of metric sketching.

Background

The question concerns the algorithmic notion of sketchability, which the paper identifies with the sketchability of traveling-salesman tours for the planar lamplighter group. The authors explain that a positive answer to the preceding Hilbert snowflake question would imply a positive answer here, but they leave sketchability itself unresolved.