This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question.
Consider two smooth plane curves $C \equiv (X_C(s),Y_C(s))$ and $S \equiv (X_S(s),Y_S(s))$ represented in arc length parametrization. Curve $C$ is asymptotic to a straight line $(a,s)$ in arc length parametrization, where $a$ is a constant. As $s\to\infty$ the curve $S$ approaches (converges) to a point in the plane, as $s\to\infty$.
Now we define moment of intertia about its center of mass, of a segment of curve $C$ between $s=s_1$ and $s = s_2$ as $$I_C(s_1,s_2) = \int_{s1}^{s_2} ((X_C(s)-X_{C_{cm}})^2 + (Y_C(s)-Y_{C_{cm}})^2) ds$$ where $(X_{C_{cm}},Y_{C_{cm}})$ is the center of mass of the segment under consideration given by $X_{C_{cm}} = \frac{1}{s_2-s_1}\int_{s_1}^{s_2}X_C(s)ds$ and $Y_{C_{cm}} = \frac{1}{s_2-s_1}\int_{s_1}^{s_2}Y_C(s)ds$. Similarly $$I_S(s_1,s_2) = \int_{s1}^{s_2} ((X_S(s)-X_{S_{cm}})^2 + (Y_S(s)-Y_{S_{cm}})^2) ds$$ where $X_{S_{cm}} = \frac{1}{s_2-s_1}\int_{s_1}^{s_2}X_S(s)ds$ and $Y_{S_{cm}} = \frac{1}{s_2-s_1}\int_{s_1}^{s_2}Y_S(s)ds$.
I'd like to prove/disprove the following statement.
Statement : Given any $s = s_1$ we can always find a sufficiently large $L$ such that for all $s_2 > L$, we have $I_C(s_1,s_2) > I_S(s_1,s_2)$.