In summary, as long as you take enough protein, extra BCAA does not promote extra muscle growth. Link to full study below.
A physiologically-significant increase in the rate of muscle protein synthesis requires adequate availability of all amino acid precursors. The source of EAAs for muscle protein synthesis in the post-absorptive state is the free intracellular pool. Intracellular free EAAs that are available for incorporation into protein are derived from muscle protein breakdown. Under normal conditions about 70% of EAAs released by muscle protein breakdown are reincorporated into muscle protein. The efficiency of reincorporation of EAAs from protein breakdown back into muscle protein can only be increased to a limited extent. For this fundamental reason, a dietary supplement of BCAAs alone cannot support an increased rate of muscle protein synthesis. The availability of the other EAAs will rapidly become rate limiting for accelerated protein synthesis. Consistent with this perspective, the few studies in human subjects have reported decreases, rather than increases, in muscle protein synthesis after intake of BCAAs. We conclude that dietary BCAA supplements alone do not promote muscle anabolism.
Source:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5568273/