  We consider the largest observed sample including all intermediate-frequency
peaked (IBL) and high-frequency peaked (HBL) flaring blazars above 100 GeV up
to redshift $z = 0.6$. We show that the best-fit regression line of the emitted
spectral indices $\Gamma_{\rm em} (z)$ is a concave parabola decreasing as $z$
increases, thereby implying a statistical correlation between the
$\{\Gamma_{\rm em} (z) \}$ distribution and $z$. This result contradicts our
expectation that such a distribution should be $z$-independent. We argue that
the above correlation does not arise from any selection bias. We show that our
expectation naturally emerges provided that axion-like particles (ALPs) are put
into the game. Moreover, ALPs can also explain why flat spectrum radio quasars
emit up to 400 GeV, in sharp contradiction with conventional physics. So, the
combination of the two very different but consistent results -- taken at face
value -- leads to a hint at an ALP with mass $m = {\cal O} (10^{-10} \, {\rm
eV})$ and two-photon coupling in the range $2.94 \times 10^{- 12} \, {\rm
GeV}^{- 1} < g_{a \gamma \gamma} < 0.66 \times 10^{- 10} \, {\rm GeV}^{- 1}$.
As a bonus, the Universe would become considerably more transparent above
energies $E \gtrsim 1 \, {\rm TeV}$ than dictated by conventional physics. Our
prediction can be checked not only by the new generation of observatories like
CTA, HAWC, GAMMA-400, LHAASO, TAIGA-HiSCORE and HERD, but also thanks to the
planned laboratory experiments ALPS II (upgraded), STAX, IAXO and with other
techniques now being developed by Avignone and collaborators.
