  Quantum interference is studied in a three-band model of pseudospin-one
fermions in the $\alpha-\mathcal{T}_3$ lattice. We derive a general formula for
magnetoconductivity that predicts a rich crossover between weak localization
(WL) and weak antilocalization (WAL) in various scenarios. Recovering the known
results for graphene ($\alpha=0$), we remarkably discover that WAL is notably
enhanced when one deviates slightly from the graphene lattice, i.e. when
$\alpha>0$, even though Berry's phase is no longer $\pi$. This is attributed to
the presence of multiple Cooperon channels. Upon further increasing $\alpha$, a
crossover to WL occurs that is maximal for the case of the Dice lattice
($\alpha=1$). Our work distinctly underscores the role of non-trivial band
topology in the localization properties of electrons confined to the
two-dimensional $\alpha-\mathcal{T}_3$ lattice.
