Quadratic Gröbner basis for toric ideals of inductively pierced codes

Let $\mathcal C$ be a neural code labeled so that neuron $i$ is added as a piercing at the $i$th step, and let $T_\mathcal C$ be its toric ideal. Let $\prec$ be the monomial order on the variables $y_c$ induced by the codeword order described in the source. Quadratic Gröbner-basis conjecture. The toric ideal $T_\mathcal C$ has a Gröbner basis consisting of quadratic polynomials with respect to the term order $\prec$. This would give a uniform Gröbner-basis property for toric ideals arising from the inductive piercing construction; the source does not state a resolution.