In the paper  [Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado,][1] page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$,  there is a map from the labelled configuration space to the section space of a $m$-sphere bundle
$$
\alpha: C(M;S^0)\to \Gamma(M;S^0)
$$
where 
$$
C(M;S^0)=\bigsqcup_{k\geq 0}F(M,k)/\Sigma_k
$$
is the disjoint union of unordered configuration spaces, and 
$$
\Gamma(M;S^0)=\bigsqcup_{q\in \mathbb{Z}} \Gamma_q(M;S^0)
$$
is the disjoint union of sections of degree $q$.

In the paper [Configuration spaces of positive and negative particles, D. McDuff,][2] Theorem 1.1, it is proved that 

(1.1) for a closed compact manifold $M$ and any fixed $n$, we can choose $k$ sufficiently large such that for any $t\geq k$, the map $\alpha$ restricted to $F(M,t)/\Sigma_t$ induces an isomorphism on $n$-th homology group
$$
(\alpha_t)_*:H_n(F(M,t)/\Sigma_t)\to H_n(\Gamma_t(M;S^0)).
$$

**Question:** Could the theorem 1.1 be strengthened to the statement that for some $k$,

$$
(\alpha_k)^*: H^*(\Gamma_k(M;S^0))\to H^*(F(M,k)/\Sigma_k)
$$
 is a ring isomorphism of cohomology rings?

  [1]: http://arxiv.org/pdf/1410.2200.pdf
  [2]: http://www.sciencedirect.com/science/article/pii/0040938375900385