Linear-type conjecture for trees and unicyclic graphs

Let $G$ be a graph, let $J_G$ be its binomial edge ideal, and say that $J_G$ is of linear type when its Rees algebra is generated by the degree-one part in the standard sense. Linear-type conjecture for trees and unicyclic graphs. If $G$ is a tree or a unicyclic graph, then

$$J_G\text{ is of linear type}.$$

The source presents this as a statement to prove or disprove after noting that the preceding classification question is not true for trees; it gives no resolution.