The bar-construction homotopy-equivalence conjecture for holomorphic bundles on the blown-up plane

Let $I\subset\mathbb C^2$ be finite. Write $\mathfrak M_I$ for the stable moduli space of holomorphic bundles associated with $I$, and let

$$\|\mathfrak B_I\|=\operatorname{BAR}\Bigl(\mathfrak M_\emptyset,\prod_{x\in I}\mathfrak M_\emptyset,\prod_{x\in I}\mathfrak M_x\Bigr)$$

be the bar construction, with its natural map $h_I\colon\|\mathfrak B_I\|\to\mathfrak M_I$ induced by Whitney sum. Bar-construction conjecture. The map

$$h_I\colon\|\mathfrak B_I\|\to\mathfrak M_I$$

is a homotopy equivalence. The result extends the established homotopy equivalence in the degree $k=1,2$ components to all components; the preceding finite-rank and infinite-charge results provide supporting evidence, but the full assertion is not resolved in the supplied text.