The idea of the conformal bootstrap is to constrain observables through symmetries, consistency and a minimal amount of physical input. Using this method, correlators of insertions on one-dimensional defects in holographic setups are particularly apt to be computed. In this journal club, I will revisit the analytic bootstrap of the paper arXiv:2004.07849 and explore the three types of constraints listed above (symmetries, consistency and physical input) and how much information can be obtained through the conformal bootstrap. I will start by presenting the idea of the conformal bootstrap and introducing the holographic defect at play, I will then explicitly bootstrap the 1st-order 4-point correlator as an example of the method. In the final section, I will present some roadblocks as well as the additional inputs that have allowed higher orders to be computed (in a paper to appear), and how more restrictive constraints may yet be uncovered.