  We present a comprehensive study on the low-lying states of neutron-rich Er,
Yb, Hf, and W isotopes across the $N=126$ shell with a multi-reference
covariant density functional theory. Beyond mean-field effects from shape
mixing and symmetry restoration on the observables that are relevant for
understanding quadrupole collectivity and underlying shell structure are
investigated. The general features of low-lying states in closed-shell nuclei
are retained in these four isotopes around $N=126$, even though the shell gap
is overall quenched by about 30\% with the beyond mean-field effects. These
effects are consistent with the previous generator-coordinate calculations
based on Gogny forces, but much smaller than that predicted by the collective
Hamiltonian calculation. It implies that the beyond mean-field effects on the
$r$-process abundances before the third peak at $A\sim195$ might be more
moderate than that found in A. Arcones and G. F. Bertsch, Phys. Rev. Lett. 108,
151101 (2012).
