= Solution
First perturb the radius consistently with the conserved string normalization. If $a_E\mapsto a_E(1+\epsilon)$ while $C_s=\rho_sa^2$ stays at its static value, the density becomes
$$
\rho_E\mapsto\rho_E(1+\epsilon)^{-2}=\rho_E(1-2\epsilon+O(\epsilon^2)).
$$
The <Friedmann equation> still has an exact cancellation between the string and curvature terms, and the <Friedmann acceleration equation> still vanishes. Thus
$$
\boxed{\text{A radius displacement at fixed }C_s\text{ moves to another static solution.}}
$$
The <homogeneous stability of a static cosmic-string universe> is neutral in this restricted sense: there is no restoring force and no exponentially growing displacement. Holding the density fixed while changing the radius would instead change $C_s$ and would not be this pure displacement perturbation.
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