Suppose $X^n$ is an orientable compact orbifold (without boundary) with stabilisers in codimesnion 2, and $\bar X^n$ is the underlying topological. We can assume moreover that $X^n$ is a quotient of a manifold $X'^n$ by an action of finite group $G$.

Is it true that $H_{n-k}(\bar X^n, \mathbb R)$ is dual to $H_k(\bar X^n,\mathbb R)$?

If not, what is a simplest contrexample? And what is the correct statement. If yes, what would be a reference?