By Proposition 3.8, $dG_{U}$ is smooth. This is compatible with the above expansion. For example, from the expansion we see the leading term involving forms of type $dx_{i}\wedge dx_{j}\wedge dy_{\beta}\wedge dy_{\gamma}$ is given by

(3.49)

$$(\frac{1}{2r}y_{\beta}A_{ij\alpha\beta}dy_{\hat{\alpha}}-\frac{1}{4}A_{ij\alpha\beta}dr\wedge dy_{\widehat{\alpha\beta}})\wedge dx_{i}\wedge dx_{j}.$$

Elementary calculation shows this vanishes.