Triple Massey products and absolute Galois groups

Let $p$ be a prime number, $F$ a field containing a root of unity of order $p$, and $G_F$ the absolute Galois group. Extending results of Hopkins, Wickelgren, Minac and Tan, we prove that the triple Massey product $H^1(G_F)^3\to H^2(G_F)$ contains $0$ whenever it is nonempty. This gives a new restriction on the possible profinite group structure of $G_F$.