The four-configuration Torelli conjecture for curves

Let $C$ be a curve of genus $\mathsf g(C)\geq 2$. An $R_4$-configuration is the configuration defined earlier in the paper, and a curve realizes an $R_4$-string when it carries such a configuration with the prescribed string. Four-configuration Torelli conjecture. For any curve $C$ of genus $\mathsf g(C)\geq 2$ there exist a string $R_4$ and an $R_4$-configuration on $C$ such that all curves $\tilde{C}$ with an $R_4$-configuration on $\tilde{C}$ realizing the $R_4$-string are Galois conjugated to $C$. Moreover, all such configurations on $C$ are also Galois conjugated. This is intended to provide a finite configuration-theoretic reconstruction of a curve up to Galois conjugacy; the surrounding discussion predicts zero-dimensional quadruple intersections, but does not establish the assertion.