Conjectural extension of Bass' theorem for locally presentable abelian categories

Let $\mathsf B$ be a locally presentable abelian category with a projective generator $P$. Let $\mathsf B_{\operatorname{proj}}$ denote the class of projective objects in $\mathsf B$. Main conjecture. The following conditions are equivalent: (1) $\mathsf B_{\operatorname{proj}}$ is covering in $\mathsf B$; (2) every direct limit of projective objects has a projective cover in $\mathsf B$; (3) every countable direct limit of copies of $P$ has a projective cover in $\mathsf B$; (4) every countable direct limit of copies of $P$ is projective in $\mathsf B$; and (5) $\mathsf B_{\operatorname{proj}}$ is closed under direct limits in $\mathsf B$. The implications (1) $\Longrightarrow$ (2) $\Longrightarrow$ (3), (5) $\Longrightarrow$ (4) $\Longrightarrow$ (3), and (5) $\Longrightarrow$ (1) are known or immediate, while the remaining displayed implications are stated to be unknown. The conjecture extends Bass' theorem from module categories to locally presentable abelian categories; some equivalences are known for categories of contramodules over topological rings.