On alternating signed permutations with the maximal number of fixed points

A conjecture by R. Stanley on a class of alternating permutations, which is proved by R. Chapman and L. Williams states that alternating permutations with the maximal number of fixed points is equidistributed with derangements. We extend this (type $A$) result to type $B$: We prove that various classes of alternating signed permutations with the maximal number of fixed points is equidistributed with certain types of derangements (of type $B$), respectively.