  The centralizer of the image of the diagonal embedding of
$U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible
representations is examined in a Schur-Weyl duality spirit. The aim is to offer
a description in terms of generators and relations. A conjecture in this
respect is offered with the centralizers presented as quotients of the
Askey-Wilson algebra. Support for the conjecture is provided by an examination
of the representations of the quotients. The conjecture is also shown to be
true in a number of cases thereby exhibiting in particular the Temperley-Lieb,
Birman-Murakami-Wenzl and one-boundary Temperley-Lieb algebras as quotients of
the Askey-Wilson algebra.
