  The BESIII Collaboration recently reported the observation of the
$a_0(980)^0-f_0(980)$ mixing in the isospin breaking decay $J/\psi\to
\eta\pi^0\phi$. In the Dalitz plot for that decay with the $\eta$ reconstructed
from two photons, there is a band around $1.4$~GeV on the $\pi^0 \phi$
distribution. In general, this peak can be due to a resonance or a kinematic
effect. In this paper, we study the effects of a set of $K^*K\bar K$ triangle
diagrams, and show that due to triangle singularities such diagrams can lead to
a peak around 1.4~GeV in the $\pi^0\phi$ invariant mass distribution. The
Dalitz plot induced by such a mechanism has a feature consistent with the
BESIII observation, namely events along the band accumulate at both ends close
to the Dalitz plot boundary. The effect of the same mechanism on the $J/\psi\to
\pi^0\pi^0\phi$ and $J/\psi \to \eta\pi^0 K^+K^-$ decays are also investigated.
We suggest to take more data for the $J/\psi\to \eta\pi^0\phi\to \eta\pi^0 K^+
K^-$ and check whether the structure around $1.4$~GeV persists for the $K^+K^-$
invariant mass away from the $\phi$ mass region. This is crucial for
understanding whether the band is due to triangle singularities or due to a
resonance. Were it the latter, the band should remain while it would not if it
is due to the former.
