Maximal sections of the unit ball of $\ell ^n_p(\mathbb C)$ for $p \gt 2$

Eskenazis, Nayar and Tkocz (2024) showed some resilience of Ball’s celebrated cube slicing theorem, namely its analogue in $\ell^n_p$ for large $p$. We show that the complex analogue, i.e. resilience of the polydisc slicing theorem proven by Oleszkiewicz and Pełczyński (2000), holds for large $p$ and small $n$, but does not hold for any $p \gt 2$ and large $n$.