= Solution
Set $X=\mathbb{CP}^4/\mathbb{CP}^1$ and $A=\mathbb{CP}^2/\mathbb{CP}^1\cong S^4$. Let $u\in H^4(A;\mathbb Z)$ be a generator and let $a\in H^4(X;\mathbb Z)$ be the class restricting to $u$. The ring computation in part (a) gives
$$
a^2\neq0\quad\text{in }H^8(X;\mathbb Z).
$$
If $r\circ\iota\simeq\operatorname{id}_A$, then <homotopy invariance of cohomology> gives $\iota^*r^*(u)=u$, hence $r^*(u)=a$. But $u^2=0$ because $H^8(S^4;\mathbb Z)=0$, whereas naturality of the <cup product> would give
$$
a^2=r^*(u)^2=r^*(u^2)=0,
$$
a contradiction. No such map $r$ exists.
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