Term-order characterization of inductively pierced neural codes

Let $\mathcal C$ be a neural code on $n$ neurons, and let $T_\mathcal C$ be its toric ideal. A code is 0- or 1-inductively pierced when it belongs to the corresponding inductively pierced class. For each $n$, there exists a term order on the polynomial ring defining $T_\mathcal C$ such that the reduced Gröbner basis of $T_\mathcal C$ contains only binomials of degree at most $2$ if and only if $\mathcal C$ is 0- or 1-inductively pierced. This conjecture proposes a Gröbner-basis criterion extending the known characterization for codes on three neurons; whether such a term order exists for every $n$ is left open in the source.