This is a classical problem with a simple solution. Make a lever-arm arrangement that is moved by the material, and sense on the arm rather than on the material itself.
For example if the web moves vertically downwards, then the lever-arm can be like a small pivoted see-saw with a bit of weight on one side, so that it will naturally rest in a horisontal position. When the web passes by it forces the see-saw to pivot down on the other side. When the web disappears, the see-saw falls back to its original position.
Then use a regular prox to sense on the see-saw.
If the web moves upwards, or horisontally, you have to change the position of the weight and maybe add an angle to the see-saw. But the principle is the same. You get the idea.
In this way you can create a lever-arm that is designed to activate on the particular kind of material, wether it be super-heavy bulk, or like in your case very light and flimsy.