The 5th and 6th order bows are very wide compared to the primary and secondary. They are made by rays that are closer to the edges of the droplets. These glancing rays are more widely colour separated when they enter and leave the drops.

The outer red rim of the 5th order bow, produced by rays emerging after five internal reflections, has a radius of 52.9º, slightly greater than that the red inner rim of the secondary bow.
 
The 5th order bow's might therefore have a chance of being photographed. Its green and blue sections fall within Alexander's dark band between the primary and secondary. With the right combination of sky lighting and perhaps polarising filters it might be photographed. Try it!

The above was written several years before it was finally positively identified in a photograph.

Harald Edens saw the same opportunity and captured part of the 5th order bow on August 8, 2012. See the image and read full details here.