An update on adding a frame of eggs is:
3/4/16 There were no Q cells on the test frame (just large larvae and sb) and no signs of a Q, eggs, or any sb in the brood box or the supers. There were masses of workers and occasional drones ( I do not think there was a swarm in July or recently). The brood frames had no brood but did have pollen and nectar, largely in the top half of brood frames. Some central brood frames had polished worker cells. Supers had capped honey and nectar).
Is it likely that in this crowded hive that: a Q (if present) has not been fertilised but is still signalling "Q substance" to the workers?; a fertile Q is present but is not laying because the hive does not need more brood to support yet?

I have an apidea with a laying Q and need to decide whether to put her into a poly nuc to over-winter or wait until I am sure about my problem hive. The least risky option seems to be to start the poly nuc. What odds would you give for each option?