  In this paper we prove an initial data rigidity result \`a la Eichmair,
Galloway and Mendes (arXiv:2009.09527) using Dirac operator techniques. It
applies to initial data sets on spin bands that satisfy the dominant energy
condition, a boundary condition for the future null expansion scalar and the
$\hat{A}$-obstruction for positive scalar curvature on one of the boundary
pieces. Interestingly, these bands turn out to carry lightlike imaginary
$W$-Killing spinors, which are connected to Lorentzian special holonomy and
moduli spaces of Ricci-flat metrics. We also obtain slight generalizations of
known rigidity results on Riemannian bands.
