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.

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.