A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters

Question

Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$?

(This problem of Aleksander Pełczyński was written on 9.08.2015 by Michał Wojciechowski on page 15 of Volume 0 of the Lviv Scottish Book).

The prize for a solution: A dinner in "Szkocka".