There are no unconditional Schauder frames of exponentials in $L^p$ spaces, $p \ne 2$

We proved recently that there are no unconditional Schauder frames of exponentials in the space $L^p(Ω)$, $1 \le p 2$, we proved that the same holds provided that $Ω$ has nonempty interior, but the general case remained open. In t