Triangle moment conjecture for ERGMs in the phase-coexistence regime

Let $\mu$ be an ERGM measure on graphs with vertex set $[n]$, and let $e_1,e_2,e_3$ be edges spanning a triangle in $K_n$. Let $e$ be an arbitrary edge. Triangle moment conjecture. Even in the phase-coexistence regime, one has

$$\mleft|\mathbb{E}_\mu[X(e_1)X(e_2)X(e_3)]-\mathbb{E}_\mu[X(e)]^3\mright|\lesssim \frac{1}{n}.$$

This would extend the bound proved for forests to graphs containing a cycle; the source notes that it can be proved under an FKG assumption but expects it to remain true in phase coexistence, where $|M_{\boldsymbol\beta}|>1$.