Yahiro for discussions. The second author would like to express his gratitude for his supervisor Takuro Mochizuki. He is also grateful Claus Hertling and Claude Sabbah for their advise, encouragement, and kindness.

The first author was supported by JSPS Grant-in-Aid for Scientific Research numbers 23224002 and 15H02054. The second author was supported by Grant-in-Aid for JSPS Research Fellow number 16J02453 and the Kyoto Top Global University Project (KTGU).

Preliminary
===========

Stokes filtrations and Stokes data {#stofil}
----------------------------------

Recall the definition of Stokes filtrations on local systems over $S^1$ in the sense of [@herexa] (see also [@sabint]). Let $\bm{k}$ be a field. Let ${\sf Loc}_{\bm{k}}:={\sf Loc}_{\bm{k}}(S^1)$ denote the category of (finite rank) $\bm{k}$-local systems on $S^1$. For $\theta\in S^1=\mathbb{R}/2\pi\mathbb{Z}$, let $\leqslant_\theta$ be the partial order on $\mathbb{C}$ defined by the following relation$:$ $$\begin{aligned}
\label{order-c}
                  c\leqslant_\theta c'
                  \Leftrightarrow 
                  \Re e^{-\mathrm{i}\theta}c < \Re e^{-\mathrm{i}\theta}c'\ \ 
                  \text{or}\ \ 
                   c=c'.
                 \end{aligned}$$ We also set $c<_\theta c'$ if and only if $c\leqslant_\theta c'$ and $c\neq c'$.

\[def stokes\] Let $\L$ be a ${\bm k}$-local system on $S^1$. A family $\L_\bullet$ of subsheaves $\L_{\leqslant c}\subset \L\ ({c\in {\mathbb{C}}})$ is called a Stokes filtration if the following conditions are satisfied$:$

1.  \[first condition of stokes structure\] For each 