radiation thermodynamics paradox

This question is concerned with a thermodynamic paradox for radiating bodies and radiation in a cavity of a specific shape.

Consider two nested shells that are axisymmetric ellipsoids with the same two foci, A and B, as shown in the figure (line AB is the axis of symmetry). Cut the system along the vertical plane of symmetry and remove the right side of the outer shell, and remove the left half of the inner shell. Then connect the two halves with vertical surface, as shown in the figure, to make it a continuous enclosure. The result is a figure of rotation shown by the thick black line in the figure.

Next, make the inner surface of it a perfect mirror. The property of such a cavity is that each ray emitted from point B comes to point A; but not each ray emitted from point A comes to point B - some rays emitted from A (shown in blue) come back to A.

Now, put two small black bodies (say, two spheres of some small radius) at points A and B. Thermodynamic equilibrium requires that eventually the temperatures of the two spheres equilibrate. However, according to the geometric properties of this cavity, all energy emitted from B comes to A but only a fraction of energy emitted from A comes to B; so the equality of temperatures is not consistent with balance of emitted and absorbed power. How to resolve this paradox?