Let $c$ be a complex sequence $c: \mathbb{Z} \to \mathbb{C}$ and define $d: \mathbb{Z} \to \mathbb{C}$ such that $d(n) = \overline{c(-n)}$. Suppose they satisfy for all $n \in \mathbb{Z}$: \begin{align*} \delta_0(n) = (c \ast d)(n) = \sum_{m \in \mathbb{Z}} c_m \overline{c_{m-n}} , \end{align*} where $\delta_m(n) = 1 \iff n = m$ is the Kronecker delta. Note that the above sum is meant to be slightly different from the normal convolution.
Can we now conclude that $c = \delta_m$ for some $m \in \mathbb{Z}$? I know for general $c$ and $d$ this does not hold, but in this situation I can't think of a counter example, and since the sequences are strongly related I would intuitively assume this to be true.