Conjecture on linear type and symbolic Rees algebras of binomial edge ideals

Let $G$ be a finite simple graph, let $J_G$ denote its binomial edge ideal, and let $C_n$ denote the cycle graph on $n$ vertices. Let $\mathcal{R}(J_G)$ be the Rees algebra of $J_G$, and let $\mathcal{R}_s(J_{C_n})$ denote the symbolic Rees algebra of $J_{C_n}$. Conjecture. If $G$ is a tree or a unicyclic graph, then $J_G$ is of linear type. Moreover,

$$\mathcal{R}_s(J_{C_n})=\mathcal{R}(J_{C_n}).$$

These claims arise from experimental evidence in the study of binomial edge ideals and their Rees algebras. The surrounding discussion also asks for a classification of all bipartite graphs whose binomial edge ideals are of linear type; the conjectured assertions provide two proposed classes of examples and a symbolic-Rees-algebra equality, with no resolution supplied here.