What size board do you have going across the bottom of the nests, how thick? 3/4" thick? 1-1/2"? That's your perch. Mine have no problems hopping/flying up more than twice your 16" and they use 1/2" thick plywood to land on. No perches, just the plywood that forms the nests. They just sort of land on it and step on in in one motion.
I don't for a minute think that perches are the problem. Those pullets are 17 weeks old. Mine show no interest at all in the nests until they are about ready to lay. About a week before they lay some start to look for a good place that will make a nest. That's when they scratch around in there, to check them out. Your problem is that they are not ready to lay.
I can't tell how high that board across the bottom of the nests are. It may or not be high enough to keep them from scratching out bedding, fake eggs, or real eggs. If you start to find bedding on the coop floor that's a good sign it isn't high enough. Those may be fine the way they are. Time will tell.
With that kind of nest, if you feel you need to put a perch, just fasten a 2x4 across the bottom. You can use it to raise the lip a bit. Mine sleep on a 2x4 narrow side up. That gives you a 1-1/2"wide perch, plenty for them to land on.