The Gröbner basis characterization of 0- and 1-inductively pierced codes

Let $\mathcal{C}$ be a code on $n$ neurons, and let $I_{\mathcal{C}}$ be its toric ideal. A reduced Gröbner basis is the unique reduced Gröbner basis of $I_{\mathcal{C}}$ with respect to a chosen monomial order.

The Gröbner basis characterization. For each $n$, there exists a monomial order such that $\mathcal{C}$ is $0$- or $1$-inductively pierced if and only if the reduced Gröbner basis of $I_{\mathcal{C}}$ contains only binomials of degree at most $2$.

This conjecture seeks a Gröbner-theoretic characterization of low-order inductive piercing for combinatorial neural codes, extending the stated result for codes on three neurons. Its resolution status is not established by the supplied source context.