Fermi liquid transport properties without quasi-particles

Today I encountered the following apparent puzzle. Suppose one has system with a self energy which is the sum of an impurity term and a marginal Fermi liquid self energy. One consequence is that the real part of the self energy is logarithmically divergent at low temperatures. Consequently, the quasi-particle weight vanishes for energies at the chemical potential.
If transport properties (such as the dc conductivity and thermal conductivity) are calculated from bubble diagrams ignoring vertex corrections then it seems the resulting expression only depends on the imaginary part (and not the real part) of the self energy. Consequently, at low temperatures the transport is dominated by impurity scattering and universal Fermi liquid properties such the Wiedemann-Franz law (and the Lorenz ratio) are obeyed.
Thus it seems one can have traditional Fermi liquid signatures without quasi-particles!

This was all stimulated by reading a nice 2002, PRL Heat Transport in a Strongly Overdoped Cuprate: Fermi liquid and a Pure d-wave BCS Superconductor. They observe that the Lorenz ratio has its universal value (to within about 1 %). I was wondering whether these observations at low temperatures had implications for recent work I did with Jure Kokalj, Consistent description of the metallic phase of overdoped cuprate superconductors as an anisotropic marginal Ferm liquid. My current view is that these experiments cannot be used to rule out a marginal Fermi liquid contribution to the self energy, but I welcome comments.