Genericity conjecture for essential dimension of objects of amenable stacks

Let $\mathcal{X}$ be an amenable stack over $k$. Let $L$ be an extension of $k$, and let $\xi$ be an object of $\mathcal{X}(\operatorname{Spec} L)$. Here $\operatorname{ed}_k\xi$ denotes the essential dimension of $\xi$, $\operatorname{g\mspace{2mu}ed}_k\mathcal{X}$ denotes the generic essential dimension of $\mathcal{X}$, and \operatorname{R_{u}}(\mathop{\underline{\mathrm{Aut}}\nolimits}_{L}\xi) is the unipotent radical of the automorphism group of $\xi$ over $L$. Genericity conjecture. One has

\operatorname{ed}_{k}\xi \leq \operatorname{g\mspace{2mu}ed}_{k} \mathcal{X} + \dim\operatorname{R_{u}}(\mathop{\underline{\mathrm{Aut}}\nolimits_{L}\xi}).

This proposed generalization of the paper's genericity theorem would bound the essential dimension of every object by the generic essential dimension of the ambient amenable stack, together with the dimension of the unipotent radical of its automorphism group. The source presents it as plausible on the basis of examples and does not give evidence that it has been proved or refuted.