Wouldn't restricting the height of the nest impair their ability to lay comfortably?
I use the rectangular cat litter buckets for nests with pine shavings as their nesting material. I use them as inserts for a shelf below the dropping trays, so I can have as many or as few as I need in there at a time and I can pull any that get soiled to clean them. I have two large fowl Cochins that can't (or won't) fit in the buckets, so there are always two spaces in the corners left open for them without buckets, about 14.5 inches tall. The large Cochins are the only ones who consistently use them out of my flock of 33 large fowl and 12 bantam hens. Everyone else seems to prefer the buckets and more often than not use them. I figure if it was that uncomfortable for them, they wouldn't use them at all, especially with other options available.

I've walked in on them laying in the bucket nests many times. They just stand as upright as they can with the height restriction. I've also found eggs in a lot of odd places, many of which would have been much more restricted for height than these. One of my big, fat Dorking hens found one of these buckets sitting on its side, even more restricted for head room, and was laying in it until I found out! It clearly doesn't bother them all that much.
