Now I got the bottom of the fault!
The battery indicator AND the way of recharching it : after more or less 48h of recharching without any sign of the Sylphyo functioning, it finally started with full charge on the indicator!
Moral : don't trust it!
It wasn't practical explaining the whole thing in the release notes, but you can find them in the Fingerings section of the user guide. I've highlighted the changes in the picture below.
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