Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$

Question

Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\mathbb N))$, $C(\beta\mathbb N\times\beta\mathbb N)$?

(The problem was posed by Anastasia & Leandro in August 2015 on page 16 of Volume 0 of the Lviv Scottish Book).

Answer 1

Yes, there are actually many of them, in particular, each separable C(K) space is complemented there. See the recent paper https://arxiv.org/abs/2405.19120.