Can a regular black hole be observationally distinguished from singular black holes as spinning lens partner in PSR-BH binaries?

To answer the question posed in the title, we consider a novel diagnostic, viz., the difference in the time of arrival (TOA) at the observer of two light rays that simultaneously emanate from a source behind a spinning lens and pass by either side of the lens to reach the observer. This is completely different from the usual Shapiro gravitational time delay, where only one onward light ray is reflected back to the observer. The TOA essentially samples the frame dragging caused by the spinning lens, apart from other lens parameters. Assuming a charged regular Ayón-Beato and García black hole as the spinning lens partner in some typical astrophysical pulsar black hole (PSR-BH) binaries, which provide the best laboratory for testing the TOA effect, we theoretically study how the prediction depends on the gyromagnetic ratio Q / M and how it compares with those when the role of spinning lens partner is played by the centrally singular Kerr–Newman and Kerr black holes. The numerical estimates for two illustrative binary lens systems show microsecond-level delay at the zeroth order, which should be measurable. However, the TOA predictions under thin-lens approximation are shown to differ only at third or higher orders of smallness indicating that the regular and singular black holes cannot be observationally distinguished despite significant qualitative differences existing among them.