It follows that $d\Theta_{1}$ and $\Upsilon$ are cohomologous on a tubular neighborhood of $\mathcal{P}$ in $\mathbb{L}$. One can also see this by a direct calculation. For example, by restricting to a slice with $z>0$ and $y=0$, it is clear by Lemma 3.33 we know $\Upsilon$ is cohomologous to $k_{+}\omega_{D}|_{H}$. On the other hand, by definition $d\Theta_{1}$ on this slice is given by $(1-\frac{k_{+}+k_{-}}{k_{-}-k_{+}})d\Gamma=k_{+}\omega_{D}$ (using Lemma 3.25)