Well, technically, MOSEK is solving both the primal and the dual, simultaneously, together. The only question here is which one MOSEK thinks is the primal, and which is the dual. In this case, CVX has decided to “swap” the primal and the dual problems, so what MOSEK calls the primal is actually the dual.
But again, since it’s solving both of them, there is no need to force it to solve the primal problem “instead”. Your original, primal solution is being extracted automatically from the dual solution MOSEK produces.