That is not a constructive proof consistent with CVX’s rules. The paper shows convexity, but doesn’t provide a DCP-compatible “constructive” proof of convexity.
Does the paper claim this problem can be solved by conic optimization, as for instance in CVX? If not, then perhaps you need to use a general nonlinear optimizer, for instance under YALMIP. If so, ask the authors for the conic (CVX) formulation.