In Lopez de Medrano "Involutions on manifolds", a homotopy smoothing of a Poincaré space $X$ is a homotopy equivalence $f:M^n\rightarrow X$, where $M^n$ is a smooth $n$-dim. manifold (everything is oriented and orientation-preserving). Two homotopy smoothings $f_i:M_i^n\rightarrow X$, $i=0,1$, are equivalent if there exists a diffeomorphism $\phi:M^n_0\rightarrow M^n_1$ such that $f_1\circ \phi\simeq f_0$. Denote by $hS(X)$ the set of equivalence classes of homotopy smoothings on $X$.

Now suppose $X$ is already a smooth, oriented $n$-dim. manifold. Under which conditions does $hS(X)$ actually correspond to a diffeomorphism classification, i.e. when is the homotopy condition $f_1\circ \phi\simeq f_0$ always satisfied?

Do you know a space $X$ where $hS(X)$ does not correspond to the diffeomorphism classification?