The stable D(2,n) problem

Let $G$ be a finitely presentable group and let $n\geq 0$. For a finite $3$-dimensional CW complex $X$ with cohomological dimension at most $2$ and fundamental group $\pi _{1}(X)=G$, form the wedge $X\vee (S^{2})^{n}$. The stable D(2,n) problem. The $D(2,n)$ problem holds for $G$ if $X\vee (S^{2})^{n}$ is homotopy equivalent to a $2$-dimensional CW complex. It is immediate that the original $D(2)$ problem implies $D(2,n)$, and $D(2,n)$ implies $D(2,n+1)$; the case $n=0$ is the original problem.