I'm almost certain this proof has been done before in one of those gradient divergence and curl fields classes.
To start the I'ld make some simplyfing assumptions. Reduce it to a 2d problem, the tubes are square and the light only exits the glass surface in a normal direction.
No matter how you arrange the squares or any reflective surfaces . . .
What they need is a one way mirrored glass surface. Just as CO2 keeps the IR light in the biosphere this surface would keep the white light from reentering the tube.
Bret