How could I use a warp grid to emulate this effect?

Hi all. I'm trying to find the most elegant way to warp a square to essentially mimic the curvature of the page of a book as it approaches the binding, like so (from example to the right):

I've tried using transform > warp, and then applying a variety of grids, but the amout of curve handles that become active when doing that make it largely problematic (as it begins to warp left and right as opposed to just up and down).

Is there a way to maybe "anchor" or "ignore" certain points in the warp grid and ONLY pull down on the ones im looking for in order to accomplish the above?

Any other ideas?

Thanks so much!