The current proof of the unimodality of all stable densities uses general harmonic function arguments to interpolate unimodality between the symmetric case and the totally asymmetric case. It is here shown that the proof is invalid.
"On the Unimodality of Stable Densities." Ann. Probab. 4 (6) 1006 - 1008, December, 1976. https://doi.org/10.1214/aop/1176995944