Normal rational curve completeness conjecture

Let $2\le N\le q-2$. In $\mathrm{PG}(N,q)$, a normal rational curve is a $(q+1)$-arc projectively equivalent to

$$\{(1,t,t^2,\ldots,t^N):t\in\mathbb{F}_q\}\cup\{(0,\ldots,0,1)\}.$$

A $(q+1)$-arc is complete if no further point can be added while preserving the arc property. Normal rational curve completeness conjecture. Every normal rational curve in $\mathrm{PG}(N,q)$ is a complete $(q+1)$-arc except when $q$ is even and $N\in\{2,q-2\}$, in which case one point can be added to the curve. This is a geometric formulation of the corresponding extremal-length question for MDS codes; the stated exceptions are precisely the cases in which an additional point may exist.