Let us first consider the case when $g=e$ is the identity matrix. If the argument below is correct, I am optimistic that the general case can be reduced to this one. 

Let $U$ be an open neighbourhood of the identity in $\mathcal D$. We want to calculate the local homology $U$ at $e$. We may assume that $U$ is mapped homeomorphically by the (inverse of) the exponential map into its image in the tangent space of $U(n)$, which can be identified with the space of skew-Hermitian $n\times n$ matrices. If I am not mistaken, elements of $\mathcal D$ that are close to the identity correspond under the exponential map to non-invertible skew-Hermitian matrices. So you are asking about the local homology in degree ${n^2-1}$ of the space of non-invertible skew-symmetric matrices. By Alexander duality, this is isomorphic to the reduced homology in degree $0$ of the space of invertible skew-symmetric matrices. An invertible skew-Hermitian matrix is determined by an unordered $n$-tuple of pairwise orthogonal lines in ${\mathcal C}^n$, together with an $n$-tuple of eigenvalues, which are all non-zero imaginary numbers. This means that the space is homotopy equivalent to 

$$U(n)/U(1)^n\times_{\Sigma_n} 2^n.$$

The set of path components of this space is $2^n/\Sigma_n$, which has $n+1$ elements. So the reduced homology has rank $n$, which confirms your conjecture.