  We show that some sets of quantum observables are unique up to an isometry
and have a contextuality witness that attains the same value for any initial
state. We prove that these two properties make it possible to certify any of
these sets by looking at the statistics of experiments with sequential
measurements and using any initial state of full rank, including thermal and
maximally mixed states. We prove that this ``certification with any full-rank
state'' (CFR) is possible for any quantum system of finite dimension $d \ge 3$
and is robust and experimentally useful in dimensions 3 and 4. In addition, we
prove that complete Kochen-Specker sets can be Bell self-tested if and only if
they enable CFR. This establishes a fundamental connection between these two
methods of certification, shows that both methods can be combined in the same
experiment, and opens new possibilities for certifying quantum devices.
