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

We proved recently that there are no unconditional Schauder frames of exponentials in the space $L^p(\mathrm{\Omega })$, $1\le p<2$, for any set $\mathrm{\Omega }\subset {\mathbb{R}}^d$ of finite measure. For $p>2$, we proved that the same holds provided that $\mathrm{\Omega }$ has nonempty interior, but the general case remained open. In this note we close this gap, i.e., we establish that exponentials cannot form an unconditional Schauder frame in $L^p(\mathrm{\Omega })$, $p\mathrm{\ne }2$, without any additional assumptions on the set $\mathrm{\Omega }$ of finite measure other than measurability.