First steps towards total reality of meromorphic functions

It was earlier conjectured by the second and the third authors that any rational curve $\gamma\colon\mathbb{CP}^1\to\mathbb{CP}^n$ such that the inverse images of all its flattening points lie on the real line $\mathbb{RP}^1\subset\mathbb{CP}^1$ is real algebraic up to a Möbius transformation of the image $\mathbb C\mathbb P^n$. (By a flattening point $p$ on $\gamma$ we mean a point at which the Frenet $n$-frame $(\gamma',\gamma'',\dots,\gamma^{(n)})$ is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic functions of degrees 2, 3 and several other cases.