  We prove a degree-one saving bound for the dimension of the space of
cohomological automorphic forms of fixed level and growing weight on
$\mathrm{SL}_2$ over any number field that is not totally real. In particular,
we establish a sharp bound on the growth of cuspidal Bianchi modular forms. We
transfer our problem into a question over the completed universal enveloping
algebras by applying an algebraic microlocalisation of Ardakov and Wadsley to
the completed homology. We prove finitely generated Iwasawa modules under the
microlocalisation are generic, solving the representation theoretic question by
estimating growth of Poincar\'e-Birkhoff-Witt filtrations on such modules.
