Does there exist an integer $0<i<8$ with the following property:

for any commutative unital ring $R$, there exists a compact Hausdorff space $X$ such that $KO^i(X)\approx K^i_{alg}(R)$?

P.S.: I am not sure if it matters but I do not particularly mind throwing in some finiteness hypotheses (say, $R$ is Noetherian).

Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of finite CW complex $X$ such that $KO^i(X)\approx K^i_{alg}(R)$?

Does there exist an integer $0<i<8$ with the following property:

for any commutative unital ring $R$, there exists a compact Hausdorff space $X$ such that $ko^i(X)\approx K^i_{alg}(R)$?

P.S.: I am not sure if it matters but I do not particularly mind throwing in some finiteness hypotheses (say, $R$ is Noetherian).

Does there exist an integer $0<i<8$ with the following property:

for any commutative unital ring $R$, there exists a compact Hausdorff space $X$ such that $KO^i(X)\approx K^i_{alg}(R)$?

P.S.: I am not sure if it matters but I do not particularly mind throwing in some finiteness hypotheses (say, $R$ is Noetherian).