Positivity of Sylow branching coefficients of symmetric groups

In this article we investigate the positivity of Sylow branching coefficients for symmetric groups when $p=3$. In particular, we complete the discussion begun by Giannelli and the author in arXiv:1712.02642 (J. Algebra) and developed in arXiv:1909.09446 (J. London Math. Soc.) concerning the case of odd primes.