On the cone of effective surfaces on $\overline{\mathcal{A}}_3$

We determine five extremal effective rays of the four-dimensional cone of effective surfaces on the toroidal compactification $\overline{\mathcal{A}}_3$ of the moduli space ${\mathcal{A}}_3$ of complex principally polarized abelian threefolds, and we conjecture that the cone of effective surfaces is generated by these surfaces. As the surfaces we define can be defined in any genus $g\ge 3$, we further conjecture that they generate the cone of effective surfaces on the perfect cone compactification $\mathcal A_g^{\mathrm{Perf}}$ for any $g\ge 3$.