Let $X$ be a compact Kaehler manifold of dimension $n$, and let $Y=\sum_{i\in I} Y_i$ be a snc divisor. In other words, one can find a finite trivializing cover $\left\{V_k;z_k^1,\ldots,z_k^n\right\}$ such that $$Y\cap U_k=\left\{z_k^1\cdots z_k^d=0\right\}.\tag{1}\label{1}$$ One denotes by $\omega_C$ a globally defined metric on the manifold $X_0=X\setminus \text{Supp}(Y)$ with conic singularities along the $\mathbb Q$-divisor $$\sum_{i\in I}(1-\frac{1}{m})Y_i$$ where $m\in\mathbb N^+$. By this we mean that on each trivializing chart $V_k$ of $X$, there exists $C_k>0$ such that $$C_k^{-1} \left(\sum_{j=1}^d \frac{idz^j_k\wedge d\overline z_k^j}{|z^j_k|^{2-\frac{2}{m}}}+\sum_{j=d+1}^n idz_k^j\wedge id\overline z^j_k\right)\leq\omega_C|_{V_k}\leq C_k \left(\sum_{j=1}^d \frac{idz^j_k\wedge d\overline z_k^j}{|z^j_k|^{2-\frac{2}{m}}}+\sum_{j=d+1}^n idz_k^j\wedge id\overline z^j_k\right),$$ and this does not depend on the chosen chart.
One then considers the local ramified maps $$\pi_k: U_k\rightarrow V_k,\,\,\,\,\,\,\, \pi_k(w_k^1\,\ldots,w_k^n):=((w_k^1)^m,\ldots,(w_k^d)^m,w_k^{d+1},\ldots,w_k^n)\tag{2}\label{2}.$$ Under this transformation, one can see that $\pi_k^* \omega_C$ is a smooth Kaehler metric over $U_k$ (i.e. it admits smooth extension) and quasi-isometric to the Euclidean metric.
My question is that, in the above setting, on every $U_k$, can we find some good complex normal/geodesic coordinates arond the origin $\left\{w^1_k,\cdots w^n_k\right\}$ with respect to the Kaehler metric ${\pi_k^*}{\omega_C}$, i.e., $${\pi_k^*}{\omega_C}=\sqrt{-1}\omega_{ml}d w^k_m\wedge d\overline w^k_l,\,\,\,\,\,\,\, \omega_{ml}=\delta_{ml}+O(|w|^2),$$ and, such that \eqref{1} and \eqref{2} hold at the same time?
For the existing of \eqref{1} and \eqref{2} simultaneously, one can see for example Remark 5.34 in Complex Geometry notes-Ye. However, I am not sure if/can't see how to achieve the much better "geodesic" coordinates even if there indeed exists some Kaehler metric on the local lifting. When adding the conic metric assumption, can we achieve this goal?
Any comments/ suggests/ references would be greatly appreciated. Thanks a lot.