Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete [geodesic metric spaces][3] such that:    

 $X_{n}$ is a regular$^1$ [CW-complex][4] of constant local dimension$^3$ $n$, it is of finite type$^4$, boundaryless$^2$, unbounded, uniform$^5$, and it is the $n$-skeleton of $X_{n+1}$, which is [n-connected][5]. Moreover, the distances  $d_{n}$ , $d_{n+1}$ generate the same topology on $X_{n}$ and  $\forall x,y \in X_{n} \ d_{n+1}(x,y) \le d_{n}(x,y)$.  
Finally $(X_{n},d_{n})$ is [quasi-isometric][6] to $(X_{n+1},d_{n+1})$, through the inclusion map $X_{n} \subset X_{n+1}$, and a distance $d$ on $ \bigcup{X_{n}}$ is defined (for $x, y \in X_{n_0}$) by    $d(x,y) := lim_{n (\ge n_0) \to \infty} d_{n}(x,y)$.   
      
         
*Definition* : Let $X:=\overline{\bigcup{X_{n}}}$ be the [completion][7] of the metric space $\bigcup{X_{n}}$ with $d$.  
**Question** : Is $X$ [weakly contractible][8] ?

*Remark:* Some of these conditions could be useless for a proof, and others, highly generalized.       
*Motivation*: See [here][1] for applications to *geometric group theory* and *noncommutative geometry*.

____  

$^1$Regular (for a CW complex) : the attaching maps are homeomorphism (see [this post][9]).      
$^2$Boundaryless (for a regular CW complex) : the boundary of each closed cell is contained is the union of the boundaries of other closed cells.     
$^3$Constant local dimension :  the topological dimension of all neighborhood of all point, is constant.   
$^4$Finite type : finitely many $r$-cells ending in a fixed $(r-1)$-cell.      
$^5$Uniform : For all $r$-cell $c_{1}$ and $c_{2}$, there is a neighborhood $n_{1}$ of $c_{1}$ and $n_{2}$ of $c_{1}$, such that $n_{1}$ is homeomorphic to $n_{2}$.

 


  [1]: http://math.stackexchange.com/questions/478588/a-combinatorial-total-space-for-torsion-free-finitely-generated-groups
  [2]: http://mathoverflow.net/questions/142529/a-problem-on-infinite-dimensional-metric-space/143837#143837
  [3]: http://en.wikipedia.org/wiki/Glossary_of_Riemannian_and_metric_geometry#G
  [4]: http://en.wikipedia.org/wiki/CW_complex
  [5]: http://en.wikipedia.org/wiki/N-connected
  [6]: http://en.wikipedia.org/wiki/Quasi-isometry
  [7]: http://en.wikipedia.org/wiki/Complete_space#Completion
  [8]: http://en.wikipedia.org/wiki/Weakly_contractible
  [9]: http://math.stackexchange.com/questions/479026/is-a-cw-complex-homeomorphic-to-a-regular-cw-complex