I am having a similar problem on one of my hives. The hive was started from a package in March. There were 8 frames 90% drawn out by about June 1 so I put on medium brood box ( using pierco frames/foundation). Now one month later it looks like they did not finish drawing out the deeps and have made very little progress on the mediums (maybe 40% on two frames). I believe we have a flow going - I see the bees working the raspberries and alfalfa. Also in the past three weeks I have put out an ample amount of honey comb to clean up from two cut outs. So there is honey for them to store. But the comb progress is slooooooow. :( One thing, I could not find the queen nor eggs. There is a lot of capped brood in the deep. Also there are a number of supercedure cells. So I suspect the hive is queenless.
Supercedure cells means only a temporary queenlessness, the bees are correcting what they see as a problem so let the supercede.
Do the bees slow down or stop comb production if the hive is queenless?
Not usually, bees are going to forage for nectar and pollen because without it they die, queenlessness is often a correctable problem as long as eggs are available.
Or does this have little bearing on the pace of comb production?
Being queenright or queenless is only indirectly involved in comb production, what affects comb production is that bees will only build comb under their feet so to keep a hive building comb it is necessary for the hive to have an expanding population. A hive can still have a population increase absent a queen as long as there is hatching brood.
The hive has a lot of bee - healthy bee count. There are a lot of bees on the pierco foundation in the mediums, but they just seem to be hanging out. Maybe I should spray the foundation with syrup?
Bees do not like to work plastic. To get them to work plastic it needs to be aired out and free of that new plastic smell. Adding coat of wax in plastic foundation is also a help. Spraying the foundation with simple syrup is another help.
Spray the syrup on so the bees have to work the plastic to get the syrup, that should get them started. If not then add a layer of wax and repeat the syrup spray.
I've seen hives that would rather abscond than work plastic.