Hardness of Double Threading with 0/1 turn costs

Determine whether Double Threading under the turn-cost metric with turn costs restricted to the set {0,1} is NP-hard.

Background

Double Threading asks for a minimum-turn threading that traverses each edge at most twice. The paper proves Double Threading is NP-complete for graphs of maximum degree 4 (with general turn costs) and provides a polynomial-time algorithm for Double Threading on graphs of maximum degree 3.

For the related variant Exactly-Double Threading, the paper shows an equivalence to Traveling Salesperson and derives polynomial-time solvability when all turn costs are in {0,1}. However, for Double Threading with turn costs restricted to {0,1}, the complexity status is not established; determining NP-hardness (or otherwise) remains open.