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Are you sure it is as much as 3mm that have to be lapped away?
That is a lot!

Yes, how could I forget that, test it first of course, sometimes stuff looks worse than it is.

That's the downside with naturals!
I use a carbide tipped tool to remove "toxic" stuff, a carbide or diamond burr in the dremel usually works fine to.
I don't know what to fill it with if you want...