  A major difficult problem in Galois theory is the characterization of
profinite groups which are realizable as absolute Galois groups of fields.
Recently the Kernel $n$-Unipotent Conjecture and the Vanishing $n$-Massey
Conjecture for $n\geq 3$ were formulated. These conjectures evolved in the last
forty years as a byproduct of the application of topological methods to Galois
cohomology. We show that both of these conjectures are true for odd rigid
fields. This is the first case of a significant family of fields where both of
the conjectures are verified besides fields whose Galois groups of $p$-maximal
extensions are free pro-$p$-groups. We also prove the Kernel Unipotent
Conjecture for Demushkin groups of rank 2, and establish a number of further
related results.
