Interpolation for degree 2 Veroneses of odd dimension

A classical fact is that through any $d+3$ general points in ${\mathbb{P}}_{\mathbb{C}}^d$ there exists a unique rational normal curve of degree $d$ passing through them. We generalize this by proving the following: when $n$ is odd, for any $\left(\genfrac{}{}{0px}{}{n+2}{2}\right)+n+1$ general points in ${\mathbb{P}}_{\mathbb{C}}^{\left(\genfrac{}{}{0px}{}{n+2}{2}\right)-1}$, there exist at least $2^{n(n-1)}$ degree 2 Veroneses passing through them. This makes substantial progress on a question of Aaron Landesman and Anand Patel, and extends the work of Arthur Coble.