Inspired by Tsirelson proof of the non Brownian character of Walsh Brownian motion ltration on three or more rays, we prove some results on a particular coupling of solutions to the interface SDE on a star graph, recently introduced. This coupling consists in two solutions which are independent given the driving Brownian motion. As a consequence, we deduce that if the star graph contains 3 or more rays, the argument of the solution at a xed time is independent of the driving Brownian motion.