Recall that $A(X)$, the K-theory of a connected, pointed space X, is defined as the K-theory spectrum of the ring spectrum $\Sigma^\infty\_+ \Omega X$ (or via a plethora of alternative definitions). Is it known if the homotopy type of $A(X)$ determines the homotopy type of $X$? If not, what is the best one can hope for?

Of course, since $X$ is connected the space $\Omega X$ with its loop space structure determines the homotopy type of $X$, but I am not sure if this is still true when we take $\Sigma^\infty\_+$, I am worried we get might get $X$ only up to $\Sigma^n \Omega^n $. Then there is the question of if ring spectra of this type can have the same K-theory, perhaps we should assume $X$ simply connected to get a positive answer?