Triangular Gröbner-basis conjecture for chemical reaction networks

Consider a chemical reaction network (CRN) whose steady-state ideal satisfies Assumptions 1–3: mass-action kinetics, inclusion of conservation laws, and absence of independent subnetworks. Let the system have variables $x_1,\dots,x_n$, and let $\mathbb{K}$ be the extension field of $\mathbb{Q}$ generated by the coefficients of its equations. Triangular Gröbner-basis conjecture. There exists a permutation $\sigma\in S_n$ such that, after re-labelling the variables

$$(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(n)}),$$

the reduced Gröbner basis of the steady-state ideal, computed in the lexicographic order

$$x_{\sigma(n)} > x_{\sigma(n-1)} > \dots > x_{\sigma(2)} > x_{\sigma(1)},$$

has the triangular shape

$$x_{\sigma(n)} - g_n(x_{\sigma(1)}),\; x_{\sigma(n-1)} - g_{n-1}(x_{\sigma(1)}),\; \dots,\; x_{\sigma(2)} - g_2(x_{\sigma(1)}),\; g_1(x_{\sigma(1)}),$$

where $g_1\in\mathbb{K}[x_{\sigma(1)}]$ and, for every $j\geq 2$, $g_j\in\mathbb{K}[x_{\sigma(1)}]$. Thus the first polynomial is univariate in $x_{\sigma(1)}$, and each remaining polynomial is linear in its corresponding variable with coefficients polynomial in $x_{\sigma(1)}$. The conjecture is motivated by numerical searches that found no CRN satisfying Assumptions 1–3 while violating the stated triangularity; the supplied context also says that triangularity is proved under a strengthened set of hypotheses, leaving the conjecture itself unresolved.