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.
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