  We prove that Vop\v{e}nka's Principle implies that for every class
$\mathfrak{X}$ of modules over any ring, the class of
\textbf{$\boldsymbol{\mathfrak{X}}$-Gorenstein Projective modules}
(\textbf{$\boldsymbol{\mathfrak{X}}$-$\boldsymbol{\mathcal{GP}}$}) is a special
precovering class. In particular, it is not possible to prove (unless
Vop\v{e}nka's Principle is inconsistent) that there is a ring over which the
\textbf{Ding Projectives} ($\boldsymbol{\mathcal{DP}}$) or the
\textbf{Gorenstein Projectives} ($\boldsymbol{\mathcal{GP}}$) do not form a
precovering class (\v{S}aroch previously obtained this result for the class
$\mathcal{GP}$, using different methods). The key innovation is a new
"top-down" characterization of \emph{deconstructibility}, which is a well-known
sufficient condition for a class to be precovering. We also prove that
Vop\v{e}nka's Principle implies, in some sense, the maximum possible amount of
deconstructibility in module categories.
