Ideal Generation Conjecture for 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. Let $d$ be the Hilbert regularity of $Z$, so the minimal generators of $I(Z)$ occur in degrees $d$ and $d+1$. Ideal Generation Conjecture. There is a Zariski dense open subset $U_{\rm igc}\subseteq(\mathbb{P}^n)^r$ such that, for every $Z\in U_{\rm igc}$, the number of minimal generators of $I(Z)$ in degree $d+1$ is

$$\max\{0,h_S(d+1)-r-(n+1)(h_S(d)-r)\}.$$

Equivalently, the multiplication map $\mu_1:I(Z)_d\otimes S_1\to I(Z)_{d+1}$ has maximal rank. The conjecture is known in the cases cited in the paper, including $n\leq4$ and certain large-$r$ ranges, but remains open in general.