Higher Castelnuovo conjecture for points in symmetric position

Let $\Gamma$ be a set of $d$ points in symmetric position in ${\bf P}^{n-1}$, where $1\leq m\leq n-3$. Suppose that

$$d\geq 2n+2m-1$$

and

$$h_\Gamma(2)=2n+m-2.$$

Higher Castelnuovo conjecture. Then $\Gamma$ lies on a curve $D$ of degree at most $n+m-2$. This is related to the Eisenbud–Harris conjecture for curves: the cases $m=1$ and $m=2$ are respectively Castelnuovo's lemma and the Eisenbud–Harris lemma, while the paper establishes the cases $m=3$ for $n\geq6$ and $m=4$ for $n\geq8$.