  Various classes of exotic singularity models have been studied as possible
mimic models for the observed recent acceleration of the universe. Here we
further study one of these classes and, under the assumption that they are
phenomenological toy models for the behavior of an underlying scalar field
which also couples to the electromagnetic sector of the theory, obtain the
corresponding behavior of the fine-structure constant $\alpha$ for particular
choices of model parameters that have been previously shown to be in reasonable
agreement with cosmological observations. We then compare this predicted
behavior with available measurements of $\alpha$, thus constraining this
putative coupling to electromagnetism. We find that values of the coupling
which would provide a good fit to spectroscopic measurements of $\alpha$ are in
more than three-sigma tension with local atomic clock bounds. Future
measurements by ESPRESSO and ELT-HIRES will provide a definitive test of these
models.
