A cubical Rips construction

Given a finitely presented group $Q$ and a compact special cube complex $X$ with non-elementary hyperbolic fundamental group, we produce a non-elementary, torsion-free, cocompactly cubulated hyperbolic group $\mathrm{\Gamma }$ that surjects onto $Q$, with kernel isomorphic to a quotient of $G={\pi }_{1}X$ and such that $max\{cd(G),2\}\ge cd(\mathrm{\Gamma })\ge cd(G)-1$.