Triangulation conjecture for clean intersections

Let $M$ be a smooth compact manifold and let $(M_i)_{i \in I}$ be a finite family of closed subspaces of $M$ that intersects cleanly. A family intersects cleanly when, locally, its members are modeled by a family of closed smooth submanifolds whose every subfamily intersection is a smooth submanifold. The pair

$$\left(M, \underset{i \in I}{\bigcup}M_i\right)$$

is a finite relative CW-complex. Triangulation conjecture for clean intersections. This asserts that cleanly intersecting subspaces admit a finite relative CW structure, extending the familiar triangulation behavior of smooth manifolds and related geometric families. The source provides examples of clean intersections, including affine subvarieties and finite families of closed subgroups of a Lie group, but gives no resolution of the conjecture.