  In a semigroup $S$ with fixed $c\in S$, one can construct a new semigroup
$(S,\cdot_c)$ called a \emph{variant} by defining $x\cdot_c y:=xcy$. Elements
$a,b\in S$ are \emph{primarily conjugate} if there exist $x,y\in S^1$ such that
$a=xy, b=yx$. This coincides with the usual conjugacy in groups, but is not
transitive in general semigroups. Ara\'{u}jo \emph{et al.} proved that
transitivity holds in a variety $\mathcal{W}$ of epigroups containing all
completely regular semigroups and their variants, and asked if transitivity
holds for all variants of semigroups in $\mathcal{W}$. We answer this
affirmatively as part of a study of varieties and variants of epigroups.
