Degree conjecture for the constant-reciprocation simplified model

Let $I_{\tilde{\mathcal C}_n}$ be the toric ideal of the simplified directed random graph model with constant reciprocation parameter $\rho$. The model is homogeneous, and its Markov basis consists of binomial relations among the edge-state variables $p_{ij}(a,b)$.

Degree conjecture for the constant-reciprocation model. The ideal $I_{\tilde{\mathcal C}_n}$ is generated in degrees $2$, $3$, and $4$, and the binomials are of the forms described for the cases $n=3$ and $n\geq4$ immediately preceding the conjecture.

The source explicitly says that Gröbner bases and Markov bases in this case remain an open problem, so the asserted degree bound and form of generators are unresolved.