  Inter-site interactions in polar lattice gases may result, due to
Hilbert-space fragmentation, in a lack of ergodicity even in absence of
disorder. We show that the inter-site interaction in a one-dimensional dipolar
gas in an optical lattice departs from the usually considered $1/r^3$
dependence, acquiring a universal form that depends on the transversal
confinement and the lattice depth. Due to the crucial role played by the
nearest- and next-to-nearest neighbors, the Hilbert-space fragmentation and
particle dynamics are very similar to that of a power-law model
$1/r^{\beta_{\mathrm{eff}}<3}$, where $\beta_{\mathrm{eff}}$ is experimentally
controllable by properly tailoring the transversal confinement. Our results are
of direct experimental relevance for experiments on dipolar gases in optical
lattices, and show that the particle dynamics may be remarkably different if
the quasi-1D lattice model is realized in a strong 3D lattice, or by means of a
strong transversal harmonic confinement.
