  The possible existence of a "bubble" structure in the proton density of
$^{34}$Si has recently attracted a lot of research interest. To examine the
existence of the "bubble" structure in low-lying states, we establish a
relativistic version of configuration mixing of both particle number and
angular momentum projected quadrupole deformed mean-field states and apply this
state-of-the-art beyond relativistic mean-field method to study the density
distribution of the low-lying states in $^{34}$Si. An excellent agreement with
the data of low-spin spectrum and electric multipole transition strengths is
achieved without introducing any parameters. We find that the central
depression in the proton density is quenched by dynamic quadrupole shape
fluctuation, but not as significantly as what has been found in a beyond
non-relativistic mean-field study. Our results suggest that the existence of
proton "bubble" structure in the low-lying excited $0^+_2$ and $2^+_1$ states
is very unlikely.
