Petrakiev's conjecture on quadratic relations of points in symmetric position

Let $\Gamma\subseteq\mathbb{P}^{e}$ be a set of $d$ points in symmetric position. For nonnegative integers $k,m$, say that $\Gamma$ satisfies the property $\propa[k,m]$ if it is not contained in any variety $Y\subseteq\mathbb{P}^{e}$ of dimension $\dim\Gamma+k$ with $\deg Y\leq\operatorname{codim}Y+m$. Petrakiev's conjecture. If $1\leq m\leq e-2$, $d\geq 2e+2m+1$, and $\Gamma$ satisfies $\propa[1,m]$, then

$$\beta_{1,1}(\Gamma)\neq\binom{e}{2}+1-m.$$

The conjecture is presented as an algebraic counterpart to the Eisenbud–Harris genus conjecture, translating the geometric question into quadratic relations of finite sets of points; the source explicitly states that it remains open.