Smoothness conjecture for complete pointsets in gauged linear sigma-models

Let ${\Scr A}$ and ${\Scr B}$ be pointsets defining a gauged linear $\sigma$-model. The pair is ${\Scr B}$-complete when

$$\operatorname{Conv}\bigl(\operatorname{Cone}(\operatorname{Conv}{\Scr A})^\vee\cap M\cap H_\nu\bigr)=\operatorname{Conv}({\Scr B}),$$

and ${\Scr A}$-complete when

$$\operatorname{Conv}\bigl(\operatorname{Cone}(\operatorname{Conv}{\Scr B})^\vee\cap N\cap H_\mu\bigr)=\operatorname{Conv}({\Scr A}).$$

Smoothness conjecture. If a gauged linear $\sigma$-model is defined by pointsets ${\Scr A}$ and ${\Scr B}$ which are both ${\Scr A}$-complete and ${\Scr B}$-complete, then this model is nonsingular for generic values of the complex-structure and Kähler parameters.

The conjecture proposes that imposing both completeness conditions prevents singularities after the extremal transitions used to enlarge the pointsets. The source gives no resolution, so the claim remains open.