Reconstructing general plane quartics from their inflection lines

Let $C$ be a general plane quartic and let $\mathrm{F}\mathrm{l}(C)$ denote the configuration of inflection lines of $C$. We show that if $D$ is any plane quartic with the same configuration of inflection lines $\mathrm{F}\mathrm{l}(C)$, then the quartics $C$ and $D$ coincide.