Let $\mathcal{S}$ be the class of finitely presented torsion-free groups which occur as the subgroup of some hyperbolic group (so for every $G\in \mathcal{S}$ there exists a hyperbolic group $H$ such that $G\hookrightarrow H$). Define $\mathcal{S}_{tf}$ to be the class of finitely presented torsion-free groups which occur as the subgroup of some torsion-free hyperbolic group. Clearly $\mathcal{S}_{tf}\subseteq \mathcal{S}$.

My question is: Are $\mathcal{S}_{tf}$ and $\mathcal{S}$ equal?

That is, when studying the class of finitely presented torsion-free subgroups of hyperbolic groups $\mathcal{S}$, is anything lost by restricting to torsion-free hyperbolic groups?

Let $\mathcal{S}$ be the class of finitely presented torsion-free groups which occur as the subgroup of some hyperbolic group (so for every $G\in \mathcal{S}$ there exists a hyperbolic group $H$ such that $G\hookrightarrow H$). Define $\mathcal{S}_{\mathrm{tf}}$ to be the class of finitely presented torsion-free groups which occur as the subgroup of some torsion-free hyperbolic group. Clearly $\mathcal{S}_{\mathrm{tf}}\subseteq \mathcal{S}$.

My question is: Are $\mathcal{S}_{\mathrm{tf}}$ and $\mathcal{S}$ equal?

That is, when studying the class of finitely presented torsion-free subgroups of hyperbolic groups $\mathcal{S}$, is anything lost by restricting to torsion-free hyperbolic groups?