  Biquard-Gauduchon have shown that conformally K\"ahler, Ricci-flat, ALF toric
metrics on the complement of toric divisors are: the Taub-NUT metric with
reversed orientation, in the Kerr-Taub-bolt family or in the Chen-Teo family.
The same authors have also given a unified construction for the above families
relying on an axi-symmetric harmonic function on $\mathbb{R}^3$. In this work,
we reverse this construction and use methods from a paper of the second named
author, "Uniqueness among scalar-flat K\"ahler metrics on non-compact toric
4-manifolds", to show that all conformally K\"ahler, Ricci-flat, toric metrics
on the complement of toric divisors, under some mild assumptions on the
associated moment polytope, are among the families above. In particular all
such metrics are ALF.
