  A locally compact groupoid is said to have the weak containment property if
its full $C^*$-algebra coincides with its reduced one. This property is
strictly weaker than amenability and is known to be equivalent to amenability
for transformation groupoids relative to actions of exact discrete groups. We
believe that for general \'etale groupoids one should have the same equivalence
of the two properties under some mild exactness assumption. In this paper we
try to support this statement.
