Minimal generation conjecture for the edge-reciprocation simplified model

Let $I_{\tilde{\mathcal E}_n}$ be the toric ideal of the simplified directed random graph model with edge-specific reciprocation, and let $\mathcal G_n$ denote the binomials arising from cycles of the associated bipartite graph. Let $Q$ be the toric ideal generated by the relations from primitive closed even walks of the complete graph $K_n$.

Minimal generation conjecture for the edge-reciprocation model. For $n\geq4$, the ideal $I_{\tilde{\mathcal E}_n}$ is minimally generated by homogeneous binomials of degrees $2$ and $3$. More precisely, the degree-$2$ and degree-$3$ binomials in $\mathcal G_n$, together with the quadratic generators of $Q$, form a Markov basis for the model.

The source presents this as a consequence that would follow from the preceding minimal Markov basis conjecture; the relevant minimal Markov basis question therefore remains open.