Symmetric exchange conjecture for bounded powers of edge ideals

Let $G$ be a finite graph on $n$ vertices and let $\mathfrak{c}\in\mathbb{Z}_{>0}^n$. Let $\delta_{\mathfrak{c}}(I(G))$ be the largest integer $q$ for which the bounded power $(I(G)^q)_{\mathfrak{c}}$ is nonzero, and let $\mathcal{B}(G,\mathfrak{c})$ be the polymatroid whose bases are the minimal monomial generators of $(I(G)^{\delta_{\mathfrak{c}}(I(G))})_{\mathfrak{c}}$. Denote by $J_{\mathcal{B}(G,\mathfrak{c})}$ the toric ideal of this polymatroid.

Symmetric exchange conjecture. The toric ideal

$$J_{\mathcal{B}(G,\mathfrak{c})}$$

of the polymatroid $\mathcal{B}(G,\mathfrak{c})$ is generated by symmetric exchange binomials.

The conjecture concerns the toric ideals arising from the maximal nonzero bounded powers of edge ideals. The paper presents it as a goal of the project and gives several classes of polymatroids for which the analogous generation property is proved, but the supplied text does not state a resolution of this general claim.