Normal bundles of hyperelliptic curves and uniform propagation of semistability

Let $C$ be a smooth hyperelliptic curve of genus $g\ge2$ over an algebraically closed field of characteristic zero. We prove that the normal bundle of the embedding of $C$ in $\mathbb{P}^n$, $n\ge3$, by the complete linear series of a general line bundle of degree $n+g$ is semistable if and only if