Compactifiability conjecture for cylindrically stratified morphisms

Let $f:X\to Y$ be a cylindrically stratified morphism of stratified topological spaces. A morphism is compactifiable if it factors as a stratified open embedding followed by a proper cylindrically stratified map. Compactifiability conjecture. Every cylindrically stratified morphism of stratified topological spaces is compactifiable in this sense. If true, this would identify all cylindrically stratified morphisms with the class used to formulate the Beck–Chevalley property for constructible sheaves; the supplied text gives no resolution, so the claim remains open.