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

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).