Infinite torsion group with K(G,1) of finite type

Question

I am wondering whether any group $G$ that is torsion and has a $K(G,1)$ of finite type (i.e. there are finitely many cells in each dimension) is already finite. The condition of having a $K(G,1)$ of finite type is equivalent to saying that $G$ is finitely presented and of type $FP_{\infty}$. Note that the question whether there are finitely presented infinite torsion groups is open, see e.g. https://mathoverflow.net/questions/78410/finitely-presented-infinite-group-with-no-element-of-infinite-order.