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In a previous paper, one of us has showed that, at least in some cases, theKerr-nature of astrophysical black hole candidates is extremely difficult totest and current techniques, even in presence of excellent data not availabletoday, cannot distinguish a Kerr black hole from a Bardeen one, despite thesubstantial difference of the two backgrounds. In this paper, we investigate ifthe detection of the "shadow" of nearby super-massive black hole candidates bynear future mm/sub-mm very long baseline interferometry experiments can do thejob. More specifically, we consider the measurement of the Kerr spin parameterof the Bardeen and Hayward regular black holes from their shadow, and we thencompare the result with the estimate inferred from the K$\alpha$ iron line andfrom the frequency of the innermost stable circular orbit. For non-rotatingblack holes, the shadow approach provides different values, and therefore theKerr black hole hypothesis can potentially be tested. For near extremalobjects, all the approaches give quite similar results, and therefore it is notpossible to constrain deviations from the Kerr solution. The present workconfirms that it is definitively challenging to test this kind of metrics, evenwith future facilities. However, the detection of a source that looks like afast-rotating Kerr black hole can put meaningful constraints on the nature ofthe compact object.

Measuring the Kerr spin parameter of regular black holes from their shadow