Expected transversality conjecture for incidence loci of complete quasimaps

Let $C$ be a curve, let $\widetilde{Q}_\beta(C,X)$ be the moduli space of complete quasimaps, and for a point $p\in C$ and a linear subspace $\Lambda\subset X$ let $\mathsf{Inc}(p,\Lambda)\subset \widetilde{Q}_\beta(C,X)$ denote the corresponding proper-transform incidence locus. Let $p_1,\ldots,p_n\in C$ and $\Lambda_1,\ldots,\Lambda_n\subset X$ be general points and general linear spaces, respectively, and set

$$I=\bigcap_{i=1}^{n}\mathsf{Inc}(p_i,\Lambda_i)\subset \widetilde{Q}_\beta(C,X).$$

Expected transversality conjecture. The intersection $I$ is generically smooth and pure of codimension

$$\sum_{i=1}^{n}(r-\dim(\Lambda_i)),$$

the expected codimension. Furthermore, any general point of $I$ lies in the non-degenerate open locus ${\mathcal{M}}^\circ_\beta(C,X)$. This conjecture asserts that the blow-ups defining complete quasimaps remove the excess-intersection phenomena occurring on low bi-rank strata, so general incidence conditions recover the expected enumerative geometry.