Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in $\mathbb{R}^2$ which contains the image $f(\mathbb{R})$ of $f$?

Yes, this question seems rather elementary, but I have already spent (or lost?) too much time on this devilish problem, and I have communicated this question to sufficiently many people to know that it is far from trivial...