Pointed representable fibrations and presheaves conjecture

Let $R(*)$ be the terminal object of $\mathrm{RepOrbSpc}$, and let $\mathrm{Rep}\{\mathbf B G\}$ be the full subcategory spanned by the classifying orbispaces of finite groups. Pointed representable-fibration conjecture. The category $\mathrm{RepOrbSpc}_*$ is equivalent to the category of pointed representable fibrations over $R(*)$ and to the category of presheaves of pointed spaces on $\mathrm{Rep}\{\mathbf B G\}$. This is the pointed analogue of the proposed description of $\mathrm{RepOrbSpc}$ by representable fibrations over $R(*)$ and presheaves on classifying orbispaces.