Van Kampen obstruction criterion for simplicial maps of trees

Let $K$ be a tree and let $\varphi:K\to\mathbb R^2$ be a simplicial map. The van Kampen obstruction $v(\varphi)$ is the mod-$2$ obstruction defined from intersections of images of disjoint simplices; for polygonal lines, its vanishing is equivalent to approximability by embeddings.

Van Kampen obstruction conjecture. The van Kampen obstruction vanishes if and only if $\varphi$ is approximable by embeddings.

The corresponding equivalence is known for polygonal lines and is false for closed polygonal lines, including the standard $3$-winding. The question for simplicial maps of arbitrary trees is attributed to R. Skopenkov and Skopenkov's cited source, and remains unresolved in the supplied text.