The Leading Terms Ideals Conjecture for rational monomial orders

Let $R$ be a strongly discrete coherent ring. For $n\a0\geq 1$, let $R[X_1,\ldots,X_n]$ be a polynomial ring, and let $\prec$ be a rational monomial order on its monomials. For a finitely generated ideal $I\subseteq R[X_1,\ldots,X_n]$, write $\operatorname{LT}_\prec(I)$ for its leading term ideal. A ring is $n$-Gröbner if every finitely generated ideal of $R[X_1,\ldots,X_n]$ has a finitely generated leading term ideal for the lexicographic order $X_1\succ\cdots\succ X_n$; it is a Gröbner ring if it is $n$-Gröbner for every $n\geq 1$. The Leading Terms Ideals Conjecture. The following conditions are equivalent: (1) for every $n\geq 1$, every rational monomial order $\prec$ on $R[X_1,\ldots,X_n]$, and every finitely generated ideal $I\subseteq R[X_1,\ldots,X_n]$, the leading term ideal $\operatorname{LT}_\prec(I)$ is finitely generated; (2) $R$ is a Gröbner ring; and (3) $R$ is $1$-Gröbner. The updated conjecture restricts the order condition to rational monomial orders after the corresponding assertion for all monomial orders was shown to be false; it reduces finite generation for these orders to the lexicographic Gröbner-ring and $1$-Gröbner properties.