There are no unconditional Schauder frames of translates in $L^p(\mathbb{R})$, $1\le p\le 2$

It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space $L^p(\mathbb{R})$ for any $1\le p<\mathrm{\infty }$. To the contrary, there do exist unconditional Schauder frames of translates in $L^p(\mathbb{R})$ for every $p>2$. The existence of such a system for $1<p\le 2$, however, has remained an open problem. In this paper the problem is solved in the negative: we prove that none of the spaces $L^p(\mathbb{R})$, $1\le p\le 2$, admits an unconditional Schauder frame of translates.