Universality conjecture for stacks of fiber functors

Let $\mathcal{X}$ be a quasi-compact fibered category over $R$, and let $\mathcal{C}$ be as in the resolution-property hypothesis of the paper. Write $\operatorname{Fib}_{\mathcal{X},\mathcal{C}}$ for the stack of fiber functors and let $\mathcal{X}\to\operatorname{Fib}_{\mathcal{X},\mathcal{C}}$ be the canonical morphism. Universality conjecture. The morphism $\mathcal{X}\to\operatorname{Fib}_{\mathcal{X},\mathcal{C}}$ is universal among maps from $\mathcal{X}$ to quasi-compact fpqc stacks with quasi-affine diagonal and the resolution property: for every such stack $\mathcal{Y}$, the induced map

$$\operatorname{Hom}(\operatorname{Fib}_{\mathcal{X},\operatorname{Vect}(\mathcal{X})},\mathcal{Y})\to\operatorname{Hom}(\mathcal{X},\mathcal{Y})$$

is an equivalence. The conjecture is verified in two special cases described in the paper: when $\mathcal{X}$ is quasi-compact with quasi-affine diagonal and $\mathcal{C}$ generates $\operatorname{QCoh}(\mathcal{X})$, and when $\mathcal{X}$ is quasi-compact and quasi-separated with $\mathcal{C}=\{\mathcal{O}_{\mathcal{X}}\}$.