  We consider two families of categories. The first is the family of semisimple
quotients of H. Andersen's tilting module categories for quantum groups of Lie
type $B$ specialized at odd roots of unity. The second consists of categories
constructed from a particular family of finite-dimensional quotients of the
group algebra of Artin's braid group known as $BMW$-algebras of type $BC$. Our
main result is to show that these families coincide as braided tensor
categories using a recent theorem of Tuba and Wenzl. The morphism spaces in
these categories can be equipped with a Hermitian form, and we are able to show
that these categories are never unitary, and no braided tensor category sharing
the Grothendieck semiring common to these families is unitarizable.
