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Zeeman's conjecture in topological combinatorics states that if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.

What is the status of this conjecture as of 2015?

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These two very recent surveys state the conjecture as open:

It doesn't seem like there's been any progress since. In particular, the most important results towards the conjecture are:

(Cohen, 1975) If $K$ is a contractible $2$-polyhedron, then $K\times I^6$ is collapsible. Similarly, if $K$ is a contractible $n$-polyhedron for $n \geq 3$, then $K \times I^{2n}$ is collapsible.

(Perelman, 2003, corollary to the Poincaré conjecture) Zeeman's conjecture is true for standard $2$-polyhedra that are spines of $3$-manifolds

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