Let $L$ be a Laurent polynomial with real coefficients, i.e., $$L(z)=\sum_{j=-r}^{s}a_{j}z^{j},$$ where $r,s\in\mathbb{N}$ and $a_{j}\in\mathbb{R}$. Assume further that the set $L^{-1}(\mathbb{R})\subset\mathbb{C}\setminus\{0\}$ contains a Jordan curve (=simple closed curve).
It can be shown that there is at most one closed curve in $L^{-1}(\mathbb{R})$ and, if the curve is present, then it has $0$ in its interior (let me stress here that $0,\infty\notin L^{-1}(\mathbb{R})$). For example, if $L$ is symmetric, i.e., $r=s$ and $a_{-j}=a_{j}$, then the Jordan curve in $L^{-1}(\mathbb{R})$ is just the unit circle $\mathbb{T}$. Another (non-symmetric) example is $$L(z)=z^{-r}(1+z)^{r+s},$$ where the Jordan curve located in $L^{-1}(\mathbb{R})$ is given by the parametrization $$\gamma(t)=\frac{\sin(\frac{r}{r+s}t)}{\sin(\frac{s}{r+s}t)}e^{it}, \quad t\in(-\pi,\pi].$$ Note also that $\mathbb{R}\setminus\{0\}\subset L^{-1}(\mathbb{R})$.
So assume that $L$ with a Jordan curve $\gamma:\mathbb{T}\to\gamma(\mathbb{T})$ in $L^{-1}(\mathbb{R})$ is given. Put $$\rho_{*}:=\min_{t\in\mathbb{T}}|\gamma(t)| \quad \text{ and } \quad \rho^{*}:=\max_{t\in\mathbb{T}}|\gamma(t)|.$$ For $\rho\in(\rho_{*},\rho^{*})$, the circle $\rho\mathbb{T}$ has finitely many intersections with the jordan curve $\gamma(\mathbb{T})$ (It can be shown that if $\gamma$ is "locally a circle", then $\gamma$ is a circle, but in this case $(\rho_{*},\rho^{*})$ is empty). For $\rho\in(\rho_{*},\rho^{*})$, let me denote the angles $0=:\theta_{0}<\theta_{1}<\dots<\theta_{k+1}:=\pi$ so that $$\rho\mathbb{T}\cap\gamma(T)=\{\rho e^{\pm i \theta_{0}},\rho e^{\pm i \theta_{1}},\dots,e^{\pm i \theta_{k+1}}\}.$$
Conjecture: Let $\rho\in(\rho_{*},\rho^{*})$ and $\theta_{0}<\theta_{1}<\dots<\theta_{k+1}$ be as above. Then either $$L(\rho e^{i\theta_{0}})<L(\rho e^{i\theta_{1}})<\dots <L(\rho e^{i\theta_{k+1}})$$ or $$L(\rho e^{i\theta_{0}})>L(\rho e^{i\theta_{1}})>\dots >L(\rho e^{i\theta_{k+1}}).$$
This peculiar monotonicity property has been observed numerically. Does anybody know an analytic argument to prove/disprove the conjecture?
As an illustration, see the plots below where $$L(z)=-1/z^3+1/z^2-7/z-9 z+2 z^2-2 z^3+z^4.$$ The first figure shows the curve $L(\rho\mathbb{T})$ and the second one the Jordan curve $\gamma$ in $L^{-1}(\mathbb{R})$ and the circle $\rho\mathbb{T}$ with $\rho=0.89$. The points in colours indicate the monotonicity property.
