I had a confusion about shallow depth Haar random quantum circuits. In this paper, in Section B (related works), it is mentioned that Haar random quantum circuits form approximate $2$-designs only after $\mathcal{O}\left(n^{1/D}\right)$ depth, where $D$ is the spatial dimension (proved here).
What does this mean for the output distribution of Haar random quantum circuits that have constant and logarithmic depth? For every fixed string $z \in \{0, 1\}^{n}$, does every output probability $|\langle z| U |0^{n} \rangle|^{2}$ approximately follow a Porter-Thomas distribution, even for a shallow depth Haar-random $U$?
If so, what extra mileage does the $2$-design property give us?