On a conjecture of G. Malle and G. Navarro on nilpotent blocks

In a recent article, G. Malle and G. Navarro conjectured that the $p$-blocks of a finite group all of whose height 0 characters have the same degree are exactly the nilpotent blocks defined by M. Brou'e and L. Puig. In this paper, we check that this conjecture holds for spin-blocks of the covering group $2.{\text{A}}_n$ of the alternating group ${\text{A}}_n$, thereby solving a case excluded from the study of quasi-simple groups by Malle and Navarro.