The converse to the pure-projective characterization of elementary torsion pairs

Let $(\mathcal{T},\mathcal{F})$ be an elementary torsion pair in $\operatorname{Mod}\textrm{-}{R}$, where $R$ is a ring. The torsion class $\mathcal{T}$ is said to have enough pure projective modules if every module in $\mathcal{T}$ admits an epimorphism from a pure projective module belonging to $\mathcal{T}$. Conjecture. If $(\mathcal{T},\mathcal{F})$ is an elementary torsion pair, then $\mathcal{T}$ has enough pure projective modules.

The preceding theorem establishes the converse implication under the additional hypothesis that $\mathcal{T}$ is definable and has a pure projective preenvelope of $R_R$. The conjecture asks whether elementary torsion pairs always have enough pure projectives, without those sufficient conditions.