The leading-monomial stability conjecture for symmetric-group invariant ideals

Order the variables as

$$x_{1}\leq y_{1}\leq\cdots\leq x_{n}\leq y_{n}.$$

Let $G_{k,n}$ be the Gröbner basis for the ideal $cmathcal{I}_{k,n}$. Let $LG_{k,n}$ be the set of leading monomials of $G_{k,n}$, and let $LG_{k,n}^{\leq m}$ be the set of leading monomials of total degree at most $m$. Leading-monomial stability conjecture.

$$LG_{k,n}^{\leq m}=LG_{k,m}^{\leq m}.$$

This predicts that the leading monomials through degree $m$ stabilize once the number of variable pairs reaches $m$. It is presented as a conjectural strengthening of the preceding stability results, and the source does not establish it in general.