Pointwise global Katz–Sarnak conjecture for $\mathcal M_g$ and $\mathcal A_g$

For each prime $l$, let $v_l(t)$ denote the local factor defined from the trace distribution in $\operatorname{GSp}_{2g}(\mathbb{Z}_l)$, with $v_p(t)=1$, and let $\operatorname{ST}_g$ be the Sato–Tate trace density. Pointwise global Katz–Sarnak conjecture. For $g\ge2$,

$$\sup_{t\in \mathbb{Z}}\left|\frac{\#\mathcal M_g(\mathbb{F}_p,t)}{\#\mathcal M_g(\mathbb{F}_p)}\sqrt p- \operatorname{ST}_g(t/\sqrt p)\prod_{l}v_l(t)\right| \longrightarrow 0$$

as $p\rightarrow\infty$. For $g\ge1$,

$$\sup_{t\in \mathbb{Z}}\left|\frac{\#\mathcal A_g(\mathbb{F}_p,t)}{\#\mathcal A_g(\mathbb{F}_p)}\sqrt p- \operatorname{ST}_g(t/\sqrt p)\prod_{l}v_l(t)\right| \longrightarrow 0$$

as $p\rightarrow\infty$. These are pointwise refinements of the stated $L^1$ predictions, incorporating all local trace factors. They are conjectural; the authors suggest that their methods might prove them at least for $\mathcal A_g$, $\mathcal M_2$, and the symmetric genus-three family.