Convex-concave body in $\mathbb{R}P^3$ contains a line

We define a class of $L$-convex-concave subsets of $\mathbb{R}P^3$, where $L$ is a projective line in $\mathbb{R}P^3$. These are sets whose sections by any plane containing $L$ are convex and concavely depend on this plane. We prove a version of Arnold hypothesis for these sets, namely we prove that each such set contains a line.