Maximum deconstructibility in module categories

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.
Comment: accepted version