  We study the vanishing of four-fold Massey products in mod p Galois
cohomology. First, we describe a sufficient condition, which is simply
expressed by the vanishing of some cup-products, in direct analogy with the
work of Guillot, Min\'a\v{c} and Topaz for p=2. For local fields with enough
roots of unity, we prove that this sufficient condition is also necessary, and
we ask whether this is a general fact.
  We provide a simple splitting variety, that is, a variety which has a
rational point if and only if our sufficient condition is satisfied. It has
rational points over local fields, and so, if it satisfies a local-global
principle, then the Massey Vanishing conjecture holds for number fields with
enough roots of unity.
  At the heart of the paper is the construction of a finite group $\tilde
U_5(\mathbb{F}_p)$, which has $U_5(\mathbb{F}_p)$ as a quotient. Here
$U_n(\mathbb{F}_p)$ is the group of unipotent $n\times n$-matrices with entries
in the field $\mathbb{F}_p$ with p elements; it is classical that
$U_{n+1}(\mathbb{F}_p)$ is intimately related to n-fold Massey products.
Although $\tilde U_5(\mathbb{F}_p)$ is much larger than $U_5(\mathbb{F}_p)$,
its definition is very natural, and for our purposes, it is easier to study.
