This is equivalent to $P\ne NP$ by the result of the paper : Birget, J.-C.; Olʹshanskii, A. Yu.; Rips, E.; Sapir, M. V. Isoperimetric functions of groups and computational complexity of the word problem. Ann. of Math. (2) 156 (2002), no. 2, 467–518. 

**Update** See the discussion [here.][1] 


  [1]: https://mathoverflow.net/questions/307629/is-this-conjecture-strictly-weaker-than-p-np