The symmetric exchange generation conjecture for unicyclic graph toric ideals

Let $G$ be a unicyclic graph on $n$ vertices, and let $\mathfrak{c} \in \mathbb{Z}_{>0}^n$. Write $\operatorname{Ker}(\pi_G^{\mathfrak{c}})$ for the associated toric ideal, and call its binomial generators symmetric exchange binomials when they arise from symmetric exchanges. Symmetric exchange generation conjecture. For every such $\mathfrak{c}$, the toric ideal

$$\operatorname{Ker}(\pi_G^{\mathfrak{c}})$$

is generated by all symmetric exchange binomials of $\operatorname{Ker}(\pi_G^{\mathfrak{c}})$. This conjecture proposes a uniform generation result for toric ideals associated with unicyclic graphs; the supplied text does not state whether it has been proved or disproved.