There is a famous open problem in group theory that asks:

Does there exist an infinite finitely presented torsion group?

The general belief being that such groups exist. I would like to know whether the existence of such a group can be ruled out if we impose stronger finiteness conditions, i.e.

Is any torsion group $G$ with a $K(G,1)$ of finite type necessarily finite?

Here, finite type means that the CW-complex $K(G,1)$ has finitely many cells in any dimension. In the previous question, one could impose even stronger conditions, e.g. requiring additionally for $G$ to have finite exponent.