Density and finite-to-one conjecture for horizontal Y-mesh configurations

Let $S$ be a horizontal $Y$-pin with $D(S)\geq p+q$, and let $X_{p+q,S}$ and $\mathcal{U}_{p+q}$ be the configuration spaces appearing in the source. Let $\pi_1:X_{p+q,S}\to\mathcal{U}_{p+q}$ be the map that records the first row. A configuration is twisted if there are a fixed $n\geq1$ and a projective transformation $\phi$ such that

$$A_{i+n}^{(j)}=\phi(A_i^{(j)}).$$

Density and finite-to-one conjecture. The map $\pi_1$ has dense image. Restricted to twisted configurations, the map $\pi_1$ is finite-to-one. This predicts both generic realizability of first-row data and finite ambiguity in the twisted setting; the source gives no evidence of resolution.