Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property $$ \left\Vert\sum_{n=1}^\infty f * h_n\right\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert $$ for all $f\in L_1(G)$. My conjecture is that there exist an angle $\theta$ and a sequence of positive reals $(r_n)$ such that $$ h_n=e^{i\theta}r_n h_1 $$ Can someone confirm or disprove it?

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