Some partial results related to this open problem can be found in the very recent preprint by F. Bogomolov and C. Böhning *On uniformly rational varieties*, see [arXiv:1307.0102][1].

In the Introduction the authors claim that this question was first raised by M. Gromov in his paper 

*Oka's principle for holomorphic sections of elliptic bundles*, Journal of the American Mathematical Society **2**, Vol. 4 (1989).

  [1]: http://arxiv.org/abs/1307.0102