  We report on our progress toward the computation of the $\pi^0$, $\eta$ and
$\eta^{\prime}$ transition form factors using staggered quarks on $N_f=2+1+1$
gauge ensembles generated by the Budapest-Marseille-Wuppertal collaboration.
These form factors are essential ingredients to evaluate the pseudoscalar-pole
contributions to the hadronic light-by-light scattering in the muon $g-2$.
Preliminary results for the pseudoscalar-pole contributions are presented, at
finite lattice spacing, for all three light mesons.
