The generalized Atiyah–Jones conjecture for four-manifolds

Let $X$ be a $4$-dimensional real manifold. Let $\mathcal{M}_k$ be the moduli space of $SU(2)$ instantons on $X$ of charge $k$, let $P_k$ be a principal bundle with $c_2(P_k)=k$, let $\mathcal{B}_k$ be the space of gauge-equivalence classes of connections on $P_k$, and let $\theta_k\colon \mathcal{M}_k\rightarrow\mathcal{B}_k$ be the inclusion. Generalized Atiyah–Jones conjecture. The maps

$$(\theta_k)_t\colon H_t(\mathcal{M}_k)\rightarrow H_t(\mathcal{B}_k)$$

and

$$(\theta_k)_t\colon \pi_t(\mathcal{M}_k)\rightarrow \pi_t(\mathcal{B}_k)$$

are isomorphisms for $t\leq q=q(k)$. This generalization extends the Atiyah–Jones prediction from instantons on the sphere to arbitrary $4$-dimensional real manifolds, and the source says that it remains open in all other cases.