General validity of the Gromov–Witten formula for transversal ADE singularities

Let $Y$ be a variety with transversal $A_n$-singularities, $n\geq 2$, and let $\rho:Z\to Y$ be its crepant resolution. Let $K$ be the line bundle on the singular locus appearing in the stated Gromov–Witten formula. The formula in Theorem

is known under the assumptions that $K$ is sufficiently ample, $H^2(Z,T_Z)=0$, and $H^1(S,T_S)=0$. **General-validity conjecture.** Theorem

remains true without the hypotheses on $K$, $H^2(Z,T_Z)$, and $H^1(S,T_S)$. The text says that an outline of a proof is given, but the supplied material does not establish whether this claim was subsequently proved.