Equivalence conjecture for the two methods of determining localized representations

Let $Q$ be a singular point of the discriminant of an elliptic fibration. Method 1 and Katz–Vafa's method are two constructions of localized representations associated with $Q$. Localized-representation methods conjecture. Refinements of Method 1 and Katz–Vafa's method determine the same localized representations. The paper notes that the relevant constructions are supported by deformation and simultaneous-resolution results and later applies the methods under genericity assumptions, but the supplied text does not establish the conjecture in full generality.