Moment of inertia about center of mass of a curve that is the arc of a circle.

Moment of inertia about center of mass of a curve that is the arc of a
circle.

Let $(x(s),y(s))$ be a smooth 2-d plane curve which is an arc of a circle
of a certain radius $r$. Assume it is represented by an inelastic string
$S$ of finite length, lying in a 2-d plane. Let there be an
infinitesimally small displacement in the inelastic string so that its new
position is given by another smooth curve $(x'(s),y'(s))$. Let $$\delta =
\max(\sup_s(|x(s)-x'(s)|), \sup_s(|x(s)-x'(s)|))$$
Consider the set $K_{\epsilon}$ of all curves $(x'(s),y'(s))$ such that
their displacement $\delta \le \epsilon$. It needs to be shown that we can
consider small enough $\epsilon$ such that the curve $(x(s),y(s))$ has the
highest moment of inertia about center of mass, among all other curves
that belong to $K_{\epsilon}$.
I'd like to know how to prove this statement if its true.
PS
The moment of inertia of a curve $(x(s),y(s))$ of finite length is given
as $\int_0^s ((x(s)-x_c)^2 + (y(s)-y_c)^2) ds$. where $(x_c,y_c)$ is the
center of mass of the curve.