Bonnor beams are light beams of infinite extent and small thickness. They have an associated spacetime (related to their energy and momentum which shapes the stress-energy tensor) of which Bonnor calculated the characteristics.
If two of these beams are parallel nothing will happen. They stay parallel forever. If they are anti-parallel though, they will converge towards each other. That is their momenta are opposite. Their energy can always be made equal by putting yourself in the right inertial frame (light doesn't have intrinsic energy like a mass has an intrinsic rest mass).
Now if we place a mass stationary wrt the beam (in our frame of reference), the mass will be dragged along in the direction of the momentum of the beam. Due to the energy of the beam, the mass will gravitate towards the beam too. and the beam will gravitate towards the mass.
This is not the case for the beams. If they move parallel, they won't gravitate to one another, nor will they be dragged along (they simply can't be dragged along because they have zero momentum wrt each other). If the beams are anti-parallel, the beams will gravitate towards one another ( because of their energy) and they will be dragged in the opposite direction to their momenta. But because their relative velocities will stay the same only their energies will be affected (so not their velocity).
But why will the beams not gravitate toward one another if they are parallel? They possess energy so you should expect this to happen. But according to the theory, it doesn't happen.
Can we prove this hypothesis by looking at the distorted images of distant heavy objects in space? einstein rings form in this case. Would they be different if the light that comes to us is converging?