  We find the contribution of constituent quark loop mechanism to the branching
ratio B_{4\gamma} = \Gamma_{4\gamma}/\Gamma_{4\gamma} \sim 5.45 \cdot 10^{-16}
for the reasonable choice of constituent quark mass m \sim 280 MeV. This result
is in agreement with vector-dominance approach result obtained years ago. Thus
the main contribution arises from QED mechanism \pi^0 \to \gamma (\gamma^*) \to
\gamma (3\gamma) including light-light scattering block with electron loop.
This contribution was investigated in paper of one of us and gave B_{4\gamma}
\sim 2.6 \cdot 10^{-11}.
