In the paper **Stability in Yang-Mills Theories** (1983), Taubes puts a (topological) bound on the Hessian of the YM-action on $S^4$. He consequently conjectured:

*"The inclusion $\mathcal{M}_n\hookrightarrow\mathcal{B}_n$ induces an isomorphism of the pointed homotopy groups $\pi_k(\cdot)$ for $k\le 2|n|$."*

Here $\mathcal{M}_n$ is the moduli space of self-dual connections on a principal $SU(2)$-bundle over $S^4$ of degree $n\in\mathbb{Z}$, and $\mathcal{B}_n$ is the space of pairs $(P,A)$ where $P$ is a principal $SU(2)$-bundle over $S^4$ satisfying $n=-c_2(P\times_{SU(2)}\mathbb{C}^2)$ (*instanton number*) and $A$ is a smooth connection on $P$.

This was 30 years ago, at a time when Yang-Mills was being squeezed for useful information. I want to guess this has been proven by now, but I cannot immediately locate a proof.

**Is this still an open problem?**

**If not, where is the proof? If so, what is currently known about the problem (with references)?**