MacPherson's conjecture on the birational invariance of $L^2$ arithmetic genus

Let $X$ be a reduced, paracompact Hermitian compact complex space, meaning that its regular part $X-\operatorname{Sing}X$ carries a locally induced Hermitian metric. Define

$$\chi_{(2)}(X-\operatorname{Sing}X):=\sum_q(-1)^q\dim H^{0,q}_{(2)}(X-\operatorname{Sing}X).$$

Let $\pi:M\to X$ be any resolution of singularities, and write $\chi(M)=\sum_q(-1)^q\dim H^{0,q}(M)$.

MacPherson's conjecture.

$$\chi_{(2)}(X-\operatorname{Sing}X)=\chi(M).$$

This proposes an extension of the birational invariance of the arithmetic genus from compact complex manifolds to Hermitian compact complex spaces. The source does not state a resolution status.