Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continues map groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying spaces $BG^{\delta}\rightarrow BG$. My question is about the the induced map in homology $i_{\ast}:H_{\ast}(BG^{\delta}, \mathbb{F}_{p})\rightarrow H_{\ast}(BG, \mathbb{F}_{p}).$ If I understand well, the map $i_{\ast}$ is conjectured to be an isomorphism (for any finite field $\mathbb{F}_{p}$).

I have read a paper by Milnor about this subject and I have heard about Morel's results in this direction (unfortunately I cannot understand Morel's Talk on the subject). So I'm wondering about the current statement and results about the conjecture (if I stated it correctly).

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying spaces $BG^{\delta}\rightarrow BG$. My question is about the induced map in homology $i_{\ast}:H_{\ast}(BG^{\delta}, \mathbb{F}_{p})\rightarrow H_{\ast}(BG, \mathbb{F}_{p}).$ If I understand well, the map $i_{\ast}$ is conjectured to be an isomorphism (for any finite field $\mathbb{F}_{p}$).

I have read a paper by Milnor about this subject and I have heard about Morel's results in this direction (unfortunately I cannot understand Morel's Talk on the subject). So I'm wondering about the current statement and results about the conjecture (if I stated it correctly).