  We prove in generic situations that the lattice in a tame type induced by the
completed cohomology of a $U(3)$-arithmetic manifold is purely local, i.e.,
only depends on the Galois representation at places above $p$. This is a
generalization to $\mathrm{GL}_3$ of the lattice conjecture of Breuil. In the
process, we also prove the geometric Breuil-M\'ezard conjecture for (tamely)
potentially crystalline deformation rings with Hodge-Tate weights $(0,1,2)$ as
well as the Serre weight conjectures over an unramified field extending our
previous results. We also prove results in modular representation theory about
lattices in Deligne-Luzstig representations for the group
$\mathrm{GL}_3(\mathbb{F}_q)$.
