  Non-parametric reconstruction of the dark energy equation of state (EoS) aims
to determine the EoS as a function of redshift without invoking any particular
dark energy model, so that the resulting EoS can be free of model-induced
biases or artifacts. Without proper regularization, however, such
reconstruction is often overwhelmed by the noise of poorly constrained modes.
An intuitive regularization scheme is to assume a priori the dark energy EoS to
evolve at most slowly with time, which may be enforced by a correlation between
the EoS at different epochs. Indeed, studies that impose the correlation prior
are able to significantly reduce the uncertainties of the reconstructed EoS and
even show hints for dynamical dark energy.
  In this work, we examine the correlation prior using mock datasets of type Ia
supernovae (SNe Ia), baryonic acoustic oscillations (BAOs), age-derived Hubble
parameter, Hubble constant, and cosmic microwave background. We find that even
though the prior is designed to disfavor evolving equations of state, it can
still accommodate spurious oscillating features at high significance. Within
the 1000 mock datasets of existing observations that are generated for the
concordance cosmological model, i.e., the input dark energy EoS $w=-1$, there
are 688 (69) cases recovering an EoS that departs from $-1$ by more than
$1\sigma$ ($2\sigma$) in one or more redshift bins. The reconstructed EoS turns
up and down markedly in many cases. Moreover, inverting the signs of the
randomly assigned errors of the mock data more or less reverses the behavior of
the EoS. Spurious results occur even more frequently when idealized SN Ia and
BAO data from future surveys are included. Our tests suggest that further
studies are needed to ensure accurate reconstruction of the EoS with the
correlation prior.
