The geometry and categorical moduli space conjecture for the stack of all curves

Let $\mathcal{M}_g^{\text{all}}$ denote the stack of all proper curves of genus $g$ over an algebraically closed field $\Bbbk$ of characteristic $0$. For each $g \ge 0$, the following assertions are conjectured:

The stack of all curves conjecture.

1. $\mathcal{M}_g^{\text{all}}$ satisfies Vakil's Murphy's law.
2. $\operatorname{Pic}(\mathcal{M}_g^{\text{all}})=0$.
3. The structure morphism

$$\mathcal{M}_g^{\text{all}}\to \operatorname{Spec}\Bbbk$$

is a categorical moduli space.

These assertions describe the expected extreme singularity of the stack together with the triviality of its Picard group and the existence of a categorical moduli space. The supplied text does not indicate whether they have been proved or disproved.