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)?**



Conjecture on homotopy groups of moduli space of self-dual connections
On afterthought I now asked Taubes himself, who pointed me to the paper **The Topology of Instanton Moduli Spaces** (Boyer-Hurtubise-Mann-Milgram). This conjecture is a form of the *Atiyah-Jones Conjecture*. The paper is essentially what is known about this problem, and in particular the above conjecture is still open. Atiyah-Jones showed that $B\_n\simeq \Omega^3\_nSU(2)$, and this paper proves the partial result:  

*The (forgetful) inclusion map $\mathcal{M}\_n\to \Omega^3\_nSU(2)$ is a homotopy equivalence through dimension $\lfloor\frac{n}{2}\rfloor-2$.*


I also asked Milgram, who confirmed that there are no further (significant) results.


I also asked Hurtubise, who remembers improving the stability range from $\frac{n}{2}$ to $n$ by computing a spectral sequence differential, but cannot remember the actual reference.


