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:
Robert Kropholler, Zeeman’s collapsibility conjecture (2013)
A. P. M. Kupers, Zeeman's conjecture (2014)
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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6$\begingroup$ There is also this recent result by Adiprasito & Benedetti (2012) Subdivisions, shellability, and collapsibility of products: For any contractible complex $C$ there is an $n\ge 0$ such that $C\times I^n$ is collapsible $\endgroup$ Commented Mar 6, 2017 at 19:01