  By constructing a new unitary representation we prove the universal group
$U(F)^+$ of Burger--Mozes does not have the Howe--Moore property when $F$ is
primitive but not $2$-transitive. It is well known $U(F)^+$ does have this
property when $F$ is $2$-transitive. Along the way, we give a characterization
of the universal group, when $F$ is primitive, to have the Howe--Moore
property, and also prove $U(F)^+$ has the relative Howe--Moore property. These
two results are a consequence of a strengthening of Mautner's phenomenon for
locally compact groups acting on d-regular trees and having Tits' independence
property.
