Strongly $FP$-injective dimensions and Gorenstein projective precovers

The existence of the Gorenstein projective precovers over $R$ an arbitrary ring, as well as the completeness of the Gorenstein projective cotorsion pair $(\mathcal{G}\mathcal{P},{\mathcal{G}\mathcal{P}}^{\perp })$, are open questions. In this paper, we provide some answers to these questions and use the tool developed to confirm the Gorenstein Symmetry Conjecture, under two different situations. We also analyze situations where the finiteness of $\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{p}(R)$ implies $\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}(R)$ finite, and its counterpart.