  We test the valley-filtering capabilities of a quantum dot inscribed by
locally straining an $\alpha$-$\mathcal{T}_3$ lattice. Specifically, we
consider an out-of-plane Gaussian bump in the center of a four-terminal
configuration and calculate the generated pseudomagnetic field having an
opposite direction for electrons originating from different valleys, the
resulting valley-polarized currents, and the conductance between the injector
and collector situated opposite one another. Depending on the quantum dot's
width and width-to-height ratio, we detect different transport regimes with and
without valley filtering for both the $\alpha$-$\mathcal{T}_3$ and dice lattice
structures. In addition, we analyze the essence of the conductance resonances
with a high valley polarization in terms of related (pseudo-) Landau levels,
the spatial distribution of the local density of states, and the local current
densities. The observed local charge and current density patterns reflect the
local inversion symmetry breaking by the strain, besides the global inversion
symmetry breaking due to the scaling parameter $\alpha$. By this way we can
also filter out different sublattices.
