Saturation Gap Conjecture for chopped ideals of general points

Let $S$ be the homogeneous coordinate ring of $\mathbb{P}^n$, let $Z$ be a general set of $r$ points, and let $I_{\langle d\rangle}=\langle I(Z)_d\rangle_S$. For positive integers $n,d,r$ satisfying $r<h_S(d)-n$, define $\gamma_n(d,Z)$ as the least positive integer $e$ such that $h_{S/I_{\langle d\rangle}}(d+e)=r$. Saturation Gap Conjecture. The value $\gamma_n(d,Z)$ depends only on $n,d,r$, so that $\gamma_n(d,Z)=\gamma_n(d,r)$, and

$$\gamma_n(d,r)=\min\left\{e\in\mathbb{Z}_{>0}\mid h_S(d+e)-r\leq\sum_{k=1}^{n-3}(-1)^{k+1}h_S(d+e-kd)\binom{h_S(d)-r}{k}\right\}.$$

This conjecture concerns the first degree in which the chopped ideal has the same Hilbert function as the ideal of the points. It is presented as implied by the Expected Syzygy Conjecture, while the paper proves the latter only in specified ranges and examples; no general resolution is given here.