In aviation there is no generally accepted meaning for the term "chambered fan," so there is no way to claim in general that a chambered fan is or is not more efficient than the corresponding ducted fan.
In fluid dynamics, straight pipes are more efficient than bent ones, so the 180 degree bends in the four exhaust pipes (and between the intake grill and the fan) are less efficient than the corresponding ducted fan design, whose intake is at the top and exhaust is at the bottom.
If this chambered fan is shielded from the airflow induced by horizontal flight, then it cannot exploit translational lift, so it would be less efficient below about 50 knots. But for horizontal flight the aircraft must tilt in that direction, pushing that airflow into the circular grill intake.
Some of the thrust leaving the four pipes would be entrained into the grill intake just below, causing at best some inefficiency, and at worst a partial vortex ring state.
The simplest way to compare the thrust and efficiency of this design to the corresponding ducted fan design blowing directly downwards would be to build a small model and measure it. Radio-control model suppliers sell cheap "EDF" electric ducted fans, and the ductwork could be 3D printed.