Monin–Rana conjecture on the equations and Gröbner bases of $\overline{M}_{0,n}$

Let $\operatorname{\overline{M}}_{0,n}$ be embedded by $\phi$ in $\mathbb{P}^1\times\cdots\times\mathbb{P}^{n-3}$, with homogeneous coordinates $w_0^{(i)},\ldots,w_i^{(i)}$ on the $i$th factor. Let $I_n$ be the prime ideal defining this image scheme-theoretically, and let $J_n$ be generated by the $2\times2$ minors in Lemma 1. Set

$$B=\bigcap_{i=1}^{n-3}\langle w_0^{(i)},\ldots,w_i^{(i)}\rangle.$$

Monin–Rana's conjecture. The ideal $I_n$ is the unique $B$-saturation of $J_n$; it is minimally generated by $\binom{n-1}{d+1}$ polynomials of degree $d$ for $d=3,4,5,\ldots,n-2$, has degree $(2n-7)!!$, and its lexicographic initial monomial ideals are square-free and Cohen–Macaulay. Equivalently, for a minimal Gröbner basis $G$ of $I_n$, the number of degree-$d$ polynomials is $\binom{n-1}{d+1}$. These equations are intended to give an explicit scheme-theoretic description of $\overline{M}_{0,n}$; the authors verified the main ideal-theoretic assertion computationally for $n=5,6,7,8$, while the general case remains open.