We modify the proof of Lemma 3.5 and Cor. 3.6. We think of the corner $q$ as the origin in the upper half plane model. Without loss of generality $v_{n}$ is the $\mathbb{R}$-translation vector field of the holomorphic strip. Then the leading order asymptotic is

$$v_{k}=(a_{k1}z^{\phi_{1}/\pi-1},\ldots,a_{kn}z^{\phi_{n}/\pi-1})+O(1),\quad k=1,2,\ldots n-1,$$

and

$$v_{n}=(a_{n1}z^{\phi_{1}/\pi},\ldots,a_{nn}z^{\phi_{n}/\pi})+O(z).$$

hence

$$\Omega(v_{1},\ldots v_{n})=z^{(\sum\phi_{k})/\pi-n+1}(\det(a_{kj})+o(1)),$$

The excess vanishing order is $\geq 1-n$, where negative order stands for poles. By the index formula (23) for the ordinary linearized Cauchy-Riemann equation, we have

$$\deg q-\deg p\geq 1-n+n=1,$$

and equality forces $\Omega(v_{1},\ldots v_{n})$ to have no interior zero, no boundary zero, minimal zero at $p$, and $\det(a_{kj})\neq 0$ at $q$. The argument in Cor. 3.6 shows $v_{1},\ldots v_{n},\frac{v_{n}}{z}$ span the real vector space of first order deformations in $W^{1,2;\pi}$. In particular, the only first order deformation which decays at $q$ is the $\mathbb{R}$-translation vector field. Thus the cokernel to the ordinary linearized Cauchy-Riemann operator vanishes, and the moduli space is regular. ∎