An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$ is a tree or a unicyclic graph, then the binomial edge ideal of $G$ is of linear type. Our investigation validates this conjecture for trees. However, our study reveals that not all unicyclic graphs adhere to this conjecture.