Red-blue alternating paths

Suppose we have two simple graphs on the same vertex set. We will call one of them red, the other blue. Suppose that for $i=1,..,k$ we have $deg (v_i)\ge i$ in both graphs, where $V_k=\{v_1,\ldots,v_k\}$ is a subset of the vertices. Is it always possible to find a family of vertex disjoint paths such that

1. for $i=1,.., k$ every $v_i$ is contained in a path,
2. each path consists of vertices only from $V_k$ except for exactly one of its endpoints which must be outside of $V_k$,
3. in each path the red and blue edges are alternating?

The claim is true if $k$ is small (<6). It is also true if the red graph and the blue graph are the same. This question was brought to my attention by a few friends who could use it in one of their papers in preparation.

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