  The Hubble horizon at matter-radiation equality ($k^{-1}_{\rm{eq}}$) and the
sound horizon at the last scattering surface ($r_s(z_*)$) provides interesting
consistency check for the $\Lambda$CDM model and its extensions. It is well
known that the reduction of $r_s$ can be compensated by the increase of $H_0$,
while the same is true for the standard rulers $k_{\rm{eq}}$. Adding extra
radiational component to the early universe can reduce $k_{\rm{eq}}$. The
addition of early dark energy (EDE), however, tends to increase $k_{\rm{eq}}$.
We perform $k_{\rm{eq}}$- and $r_s$-based analyses in both the EDE model and
the Wess-Zumino Dark Radiation (WZDR) model. In the latter case we find $\Delta
H_0 = 0.4$ between the $r_s$- and $k_{\rm{eq}}$-based datasets, while in the
former case we find $\Delta H_0 = 1.2$. This result suggests that the dark
radiation scenario is more consistent in the fit of the two standard rulers
($k_{\rm{eq}}$ and $r_s$). As a forecast analyses, we fit the two models with a
mock $k_{\rm{eq}}$ prior derived from \emph{Planck} best-fit $\Lambda$CDM
model. Compared with the best-fit $H_0$ in baseline $\Lambda$CDM model, we find
$\Delta H_0 = 1.1$ for WZDR model and $\Delta H_0 = - 2.4$ for EDE model.
