Consider a one-sided ( say, internal) neighborhood $U$ of the unit circle $S$ ( containing $S$) on the plane with a choice of complex stricture $\tau$ on it.

Two such pairs $(U_1,\tau_1)$ and $(U_2,\tau_2)$ are equivalent if there are subneighborhoods $V_1\subset U_1$ and $V_2\subset U_2$ and a diffeomorphism $f: V_1\rightarrow V_2$ which is biholomorphic in the interior of $V_1$ with respect to $\tau_1, \tau_2$.

What is the moduli space of such structures?