Nice find
The Mantle/Mays card is a tough one to find in any shape and Mantle is becoming extremely desirable of late.
I don't know how much success you might have with the Mantle if you soaked it, but I surely would give it a shot if it were my card. At the very least, you have a chance to be able to get some of that paper off and just have a glue stain over a legible back. I don't think you could reduce the condition any further. Just make sure you give it ample time to dry after soaking or it will warp. If you are not familiar with soaking, it can be a pain to get the card dried properly, but you can't ruin it even if it warps, as it can always be re-soaked.
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