  This article is a contribution to the following problem: does there exist a
Polish non-archimedean group (equivalently: automorphism group of a Fraisse
limit) that is extremely amenable, and has ample generics. As Fraisse limits
whose automorphism groups are extremely amenable must be ordered, i.e.,
equipped with a linear ordering, we focus on ordered Fraisse limits. We prove
that automorphism groups of the universal ordered boron tree, and the universal
ordered poset have a comeager conjugacy class but no comeager $2$-dimensional
diagonal conjugacy class. We also provide general conditions implying that
there is no comeager conjugacy class, comeager $2$-dimensional diagonal
conjugacy class or non-meager $2$-dimensional topological similarity class in
the automorphism group of an ordered Fraisse limit. We provide a number of
applications of these results.
