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I have a few of the same board that I think would be high peripheral, but it's got these ferrous shields covering many components on both sides of the board. They don't come off easily at all, they're stubborn, sharp, and actually start to peel off the top layer of the board in places. Should I leave the shields on? Would they be high peripheral if I do?

Pictured are 3 of the same board showing the front and back with the two top boards, and the bottom board is my attempt to partially deshield it, showing sharp remnants and board peeling, but that there are good chips underneath.

I use a small sharp side cutter to remove shields like you have pictured. I cut into a corner and cut further and lift and continue cutting and twisting and lifting. That technique works fine for me. Other board members use other techniques. Working out a comfortable technique for yourself is " where it's at" in my opinion.
Removing shields will help the grade. Briefly. Metal is junk. Some is acceptable in small or known quanities. But over all remove junk. That is. "Where it's at". Now. There are posts on this site that cover junk. Also. You can learn about junk from the videos on the e-price list. Study junk in the E-scrap business. Removing it is a postive thing to do.
Your boards. I see High Peripheral. You may get Low Telecom. Again. Remove the shielding and do yourself and any board member justice for evaluations.
Here to learn like everybody else
My evaluations are my own opinion

Removing shields is almost always necessary if you want the best price for your boards, especially if they are ferrous. It removes the lowest price, and fairly heavy, metal from the board and it reveals what's actually underneath it so Boardsort (and you) don't have to guess.

It is annoying when the shields are soldered directly to the board, so if you aren't pinching pennies (hmm, should we start saying nabbing nickels instead?) you could leave them on. Know that it will affect the grade.

If they have a lid that pops off and leaves a separate thin strip of metal (I like to call it a fence ¯\_(ツ)_/¯) you can usually leave that but sometimes the fence makes the difference in grades.

Cutters work as catwhisker described; sometimes a flathead screwdriver can be pressed into the side of the shield breaking the solder and giving you a place to pry up the rest of it or sometimes the shield has holes, like yours do and you can use those to put something in to pry up on.

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Here to learn more so I can recycle more
My grades are my own opinion and not an official grade from Boardsort