Discriminants of convex curves are homeomorphic

For a given real generic curve $\gamma : S^{1}\to \mathbb{P}^{n}$ let $D_{\gamma }$ denote the ruled hypersurface in $\mathbb{P}^{n}$ consisting of all osculating subspaces to $\gamma$ of codimension 2. In this note we show that for any two convex real projective curves $\gamma _{1}:S^{1}\to \mathbb{P}^{n}$ and $\gamma _{2}:S^{1}\to \mathbb{P}^{n}$ the pairs $(\mathbb{P}^{n},D_{\gamma _{1}})$ and $(\mathbb{P}^{n},D_{\gamma _{2}})$ are homeomorphic.