Let $\cal{A}$ denote the family of all twice-differentiable simple open arcs $A$ in $\mathbb{R}^3$ satisfying the following properties:

(1) at each point of every arc $A\in\cal{A}$ the curvature of $A$ is at most $1$;

(2) no perpendicular projection of $A$ to any plane is an injection.

Question 1. What is the infimum of the length $|A|$ of $A\in\cal{A}$ ?

Question 2. Is there an arc $A_{min}$ of minimum length among all arcs in $\cal{A}$? If so, is it unique (up to isometry)?

Remark. This question is related to the notion of rope knots, see: Some questions about ideal knots