Non-formality of Galois cohomology modulo all primes

Let p be a prime number and let F be a field of characteristic different from p . We prove that there exist a field extension $L/F$ and $a,b,c,d$ in L× such that $(a,b)=(b,c)=(c,d)=0$ in Br(L)[p] but the mod p Massey product a,b,c,d is not defined over L . Thus, the strong Massey vanishing conjecture at the prime p fails for L , and the cochain differential graded ring C*(Γ L, Z/p Z) of the absolute Galois group Γ L of L is not formal. This answers a question of Positselski. As our main tool, we define a secondary obstruction that detects non-triviality of unramified torsors under tori, and which is of independent interest.