The bi-arithmetic-point characterization conjecture

Let $S_\Gamma$ be the variety under consideration, and let bi-arithmetic points and bi-arithmetic subvarieties be defined as in the paper. The bi-arithmetic-point characterization conjecture. An irreducible subvariety $W\subset S_\Gamma$ contains a dense set of bi-arithmetic points if and only if it is bi-arithmetic. This is the final conjecture proposed in the paper concerning Zariski closures of distinguished points; no resolution is supplied.