  We investigate the factor types of the extremal KMS states for the preferred
dynamics on the Toeplitz algebra and the Cuntz--Krieger algebra of a strongly
connected finite $k$-graph. For inverse temperatures above 1, all of the
extremal KMS states are of type I$_\infty$. At inverse temperature 1, there is
a dichotomy: if the $k$-graph is a simple $k$-dimensional cycle, we obtain a
finite type I factor; otherwise we obtain a type III factor, whose Connes
invariant we compute in terms of the spectral radii of the coordinate matrices
and the degrees of cycles in the graph.
