Complete intersection classification conjecture for reduced subvarieties of Veronese surfaces

Let $V_{n,d}\subseteq\mathbb P^N$ be the Veronese variety, with $d>1$, and let $\mathbb X\subseteq V_{n,d}$ be a reduced subvariety that is a complete intersection of type $(a_1,\ldots,a_r)$, where $a_1\leq\cdots\leq a_r$.

Complete intersection classification conjecture. Such a subvariety exists if and only if it is one of the following:

• $r=N$ and $a_1=\cdots=a_N=1$, for arbitrary $n,d$; then $\mathbb X$ is a reduced point.
• $r=N$, $a_1=\cdots=a_{N-1}=1$, and $a_N=2$, for arbitrary $n,d$; then $\mathbb X$ is a set of two reduced points.
• $r=N$, $a_1=\cdots=a_{N-2}=1$, $a_{N-1}=2$, and $a_N=b$, with arbitrary $n$, $d=2$, and $a\geq2$; then $\mathbb X=\mathcal C\cap H_b$, where $\mathcal C\subseteq V_{n,2}$ is a conic and $H_b$ is a degree-$b$ hypersurface.
• $r=N-1$, $a_1=\cdots=a_{N-2}=1$, $a_{N-1}=2$, with $d=2$ and arbitrary $n$; then $\mathbb X$ is a conic.

The claim is proved in the source for $n\leq2$ with arbitrary $d$, and for $n=3$, $d=2$; the general classification remains open.