The paper towel in the tub looks like its under water. Can you soak it in hot water, wring it out and crumple it up, then and place it back in the bin? The surface area of the paper towel crumpled up is what generates more humidity. Yes, the membranes in the pipped eggs may dry out and cause...
Maybe you can soak some paper towels in hot water and wring them out, then place the damp crumpled towels in that plastic container next to the eggs. It should not touch the eggs. If you have a makeshift cover for that plastic tub that would hold humidity better, but I guess the heat from the...