Zeeman's conjecture on collapsibility after subdivision

Let $K$ be a contractible $2$-complex. A barycentric subdivision is the subdivision obtained by replacing each simplex by the simplicial complex of its faces. The product $K\times \text{I}$ is formed with the interval $\text{I}$, and a complex is collapsible if it can be reduced to one of its vertices by a sequence of elementary collapses.

Zeeman's conjecture. After taking finitely many barycentric subdivisions, the product

$$K \times \text{I}$$

is collapsible.

Collapsibility is stronger than contractibility, and the conjecture remains open; the source states that it implies the three-dimensional conjecture. It is a central problem in simple homotopy theory concerning contractible $2$-complexes.