Q:

Status of Zeeman's collapsability Conjecture

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? 

A:

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

