The descriptive complexity of the set of arc-connected compact subsets of the plane

We compute the exact complexity of the set of all arc-connected compact subsets of $\boldmath R^2$, which turns out to be strictly higher than the classical $\boldmath Σ^1_1$ and $\boldmath Π^1_1$ classes of analytic and coanalytic sets, but stricly lower than the class $\boldmath Π^1_2$ which is the exact descriptive class of the set of all arc-connected compact subsets of $\boldmath R^3$.