Disjoint-arc criterion for approximability by embeddings

Let $I$ be an interval and let $\varphi:I\to\mathbb{R}^2$ be a piecewise linear path. Two maps are approximable by link maps when they can be approximated by maps with disjoint images.

Disjoint-arc criterion. The path $\varphi$ is approximable by embeddings if and only if, for every pair of disjoint arcs $I_1,I_2\subset I$, the restrictions $\varphi:I_1\to\mathbb{R}^2$ and $\varphi:I_2\to\mathbb{R}^2$ are approximable by link maps.

This criterion relates approximability by embeddings to the corresponding disjoint-image problem for every pair of disjoint subarcs. The supplied text gives no resolution status.