SCHWEITZER 2008 Problem 10

Let $V$ be the set of non-collinear pairs of vectors in $\mathbb{R}^3$ , and $H$ be the set of lines passing through the origin. Is is true that for every continuous map $f\colon V\rightarrow H$ there exists a continuous map $g\colon V\rightarrow \mathbb{R}^3\,\backslash\,\{ 0\}$ such that $g(v)\in f(v)$ for all $v\in V$ ?

(translated by Miklós Maróti)