Instantons on connected sums and the bar construction

Let $\mathfrak M_k^r X$ denote the moduli space of based $SU(r)$ instantons on a 4-manifold $X$ with second Chern class $k$ and let $\mathfrak M^rX=\coprod_k\mathfrak M_k^r X$. When $X$ and $Y$ are connected sums of projective planes we show that, for $k=1,2$, we have homotopy equivalences betwen $\mathfrak M_k^r(X\# Y)$ and the degree $k$ components of both $\text{Bar}(\mathfrak M^r X,\mathfrak M^r S^4,\mathfrak M^r Y)$ and $\text{Bar}\bigl(\mathfrak M^r S^4,(\mathfrak M^r S^4)^{n},(\mathfrak M^r\mathbb{CP}^2)^{n}\bigr)$, where $n$ equals the second Betti number of $X\#Y$. An analogous result holds in the limit when $k\to\infty$. As an application we obtain bounds to the Atiyah-Jones map in the rank-stable limit.