Expected Syzygy 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 $I(Z)$ be its vanishing ideal. For an integer $d$ that is the smallest value such that $h_S(d)\geq r$, write $I_{\langle d\rangle}=\langle I(Z)_d\rangle_S$ and let $e_0>0$ be the smallest integer such that the sum below is at least $h_S(d+e_0)-r$. Expected Syzygy Conjecture. For every $e\geq 0$,

$$h_{I_{\langle d\rangle}}(d+e)=\begin{cases}\displaystyle\sum_{k\geq 1}(-1)^{k+1}h_S(d+e-kd)\binom{h_S(d)-r}{k}&e<e_0,\\ h_S(d+e)-r&e\geq e_0. \end{cases}$$

The conjecture expresses the expectation that, generically, the degree-$d$ equations of a set of points are as independent as possible, equivalently that their syzygies are generated by Koszul syzygies as long as the containment $I_{\langle d\rangle}\subseteq I(Z)$ permits. The paper proves it in several infinite families and many computationally verified low-dimensional cases, but it is not established in general.