Weakened Minimal Resolution Conjecture for points in $\mathbb P^1\times\mathbb P^2$

Let $N\geq 2$ be an integer, and let $U\subseteq(\mathbb P^1\times\mathbb P^2)^N$ be a dense open subset. For $(P_1,P_2,\dots,P_N)\in U$, set $X=\{P_1,\dots,P_N\}$, and let $S/I_X$ be its coordinate ring with bigraded Betti numbers $\beta_{1,(i,j)}$. For a matrix $H$ indexed by $(i,j)\in\mathbb Z^2$, write

$$DH=(\Delta^C)^2(\Delta^R)^3H.$$

Let $H_X$ be the Hilbert matrix of $X$.

Weakened Minimal Resolution Conjecture. The set $X$ has a generic Hilbert matrix as in the source definition, and, for every fixed $(i,j)>(0,0)$, one has $\beta_{1,(i,j)}>0$ if and only if

$$DH_X(i,j)<0\quad\text{and}\quad DH_X(i',j')\leq 0$$

for all $(i',j')\leq(i,j)$ with $(i',j')\neq(0,0)$. Whenever this condition holds,

$$\beta_{1,(i,j)}=-DH_X(i,j).$$

This is a weakened version of the Minimal Resolution Conjecture for sufficiently general points in $\mathbb P^1\times\mathbb P^2$: it only prescribes the first Betti numbers, but that control is intended to determine enough of the resolution to establish the paper's virtual-resolution results. The parser supplies no evidence that the conjecture has been resolved, so its status remains open.