  We present preliminary lattice results for the topological susceptibility in
high-$T$ $N_f=2+1$ QCD obtained discretizing this observable via spectral
projectors on eigenmodes of the staggered operator, and we compare them with
those obtained with the standard gluonic definition. The adoption of the
spectral discretization is motivated by the large lattice artifacts affecting
the continuum scaling of the gluonic susceptibility at high $T$, related to the
choice of non-chiral fermions in the action.
