Stratified rigidity question for Scott spaces

Let $M$ be a Scott complex or quaternionic manifold with its natural stratification. Let $N$ be a stratified space and let $f:N\to M$ be a stratified homotopy equivalence. Stratified Rigidity Question. Then $f$ is stratified homotopic to a stratified homeomorphism. This is presented as a rigidity problem related to quasitoric and quoric manifolds and as the stratified analogue of the rigidity question; the source gives no resolution status.