On mutual arrangements of a plane real curve relative to an $M$-quartic with an oval-snake

An oval $O$ of a plane real algebraic quartic curve $S$ is called a snake coiling around a real curve $C_k$ of degree $k$ if $O\cup \mathbb{R}{C}_k$ is isotopic to $O^{\mathrm{\prime }}\cup \mathbb{R}{C}_k$, where $O^{\mathrm{\prime }}$ is the boundary of a thickening of the embedded segment that transversally intersects $\mathbb{R}{C}_k$ at $2k$ points. In this article we prove that in this case $\mathbb{R}{C}_k\cup \mathbb{R}S$ is isotopic to $\mathbb{R}{C}_k\cup \mathbb{R}Q$, where $Q$ is a perturbation of the doubled conic. We prove analogs of this statement for real pseudoholomorphic curves under some additional assumptions.