The following problem is listed here: http://www-personal.umich.edu/~erman/Papers/Questions2.pdf and attributed to Vistoli:

Let $\mathcal A_g$ denote the moduli stack of principally polarized abelian varieties over a field $k$, let $A_g$ denote the associated coarse moduli space, and $K(A_g)$ its function field. What is the minimal degree of a **finite** extension $L/K(A_g)$ so that there exists a commutative diagram:
$$\begin{matrix}\operatorname{Spec}L&\to&\mathcal A_g\cr\downarrow&&\downarrow&&?\cr\operatorname{Spec}K(A_g)&\to&A_g\end{matrix}$$
The paper "Essential dimension of moduli of curves and other algebraic stacks" by Brosnan--Reichstein--Vistoli--Fakhruddin is listed as a reference. The last section of this paper (by Fakhruddin) calculates the essential dimension of the stack $\mathcal A_g$. My question is: how is the notion of essential dimension of a stack related to the problem posed by Vistoli?

For reference, the **essential dimension** of a functor $F:\mathfrak F\mathfrak i\mathfrak e\mathfrak l\mathfrak d\mathfrak s_{/k}\to\mathfrak S\mathfrak e\mathfrak t\mathfrak s$ is the minimal integer $n$ such that every element of $F(K/k)$ comes from an element of $F(L/k)$ for some $L/k$ of **transcendence degree** at most $n$ over $k$ (if no such $n$ exists, then the essential dimension of $F$ is infinite). The essential dimension of a functor $F:\mathfrak F\mathfrak i\mathfrak e\mathfrak l\mathfrak d\mathfrak s_{/k}\to\mathfrak G\mathfrak r\mathfrak o\mathfrak u\mathfrak p\mathfrak o\mathfrak i\mathfrak d\mathfrak s$ is just the essential dimension of $\left|F\right|$ (where $\left|\cdot\right|:\mathfrak G\mathfrak r\mathfrak o\mathfrak u\mathfrak p\mathfrak o\mathfrak i\mathfrak d\mathfrak s\to\mathfrak S\mathfrak e\mathfrak t\mathfrak s$ denotes taking isomorphism classes).