  In this work, a unimodular random planar triangulation is constructed that
has no invariant circle packing. This disputes a problem asked in
[arXiv:1910.01614]. A natural weaker problem is the existence of
point-stationary circle packings for a graph, which are circle packings that
satisfy a certain mass transport principle. It is shown that the answer to this
weaker problem is also false. Two examples are provided with two different
approaches: Using indistinguishability and finite approximations.
