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These stones are to the softer side - like an Escher.
A Norton lapping stone would be a good start - to remove the saw marks.
However, you will quickly progress to 600 and then 1000 grit sandpaper.

I do not live in Italy and I am not a miner.
However, a google search shows that slate is mined near Imperia.
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Received my ILR today:
1. Appearance: very light grey - completely uniform in color - no defects/inclusion.
2. Surface: as advertised - lapped flat, but not smoothed.
Easily/quickly finished...