Higher Triple Massey products and symbols

Fix an arbitrary prime p. Let F be a field containing a primitive p-th root of unity, with absolute Galois group G _F , and let H ⁿ denote its mod p cohomology group H ⁿ (G _F ,Z/pZ). The triple Massey product of weight (n,k,m)∈N ³ is a partially defined, multi-valued function 〈⋅,⋅,⋅〉:H ⁿ ×H ^k ×H ^m →H ^n+k+m−1 . In this work we prove that for an arbitrary prime p, any defined 3MP of weight (n,1,m), where the first and third entries are symbols, contains zero; and that any defined 3MP of weight (1,k,1), where the middle entry is a symbol, contains zero.