How do I get rid of the hard particulate matter embedded within my window screen

I moved into a place with one very hazy window screen that is difficult to see through.

I have tried scrubbing it with a brush with soapy water and blasting it with the hose. But I cannot get rid of the hard material buildup. I even tried dipping a brush in CLR and scrubbing the screen.

When I Google how to clean a window screen, I just run across people telling me to do what I already tried. But I think the people touting these methods simply have screens with dirt in them.

This doesn't seem like simple dirt buildup. I enclosed a close up photo of what it looks like.

I bet if I poked all of the holes individually with a toothpick I could clean it, but that would take eons. Any advice? Thanks.