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

By smoothness of $\tau$ on $U$ we mean that corresponding operator of almost-complex structure $J$( defined in the interior of $U$) as a matrix with respect to the coordinate basis on the plane, has smooth extension to $S$.

Two such pairs $(U_1,\tau_1)$ and $(U_2,\tau_2)$ are equivalent if there are subneighborhoods $V_1\subset U_1$ and $V_2\subset U_2$ ( both containing $S$) 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?