AFAIK ledger will only support the fork that will dominate and I think that's BCH ABC.

https://support.ledgerwallet.com/hc/en-us/articles/360012036813-Preparing-for-Bitcoin-Cash-hard-fork

But now, wait for their further announcement and follow them on twitter.

Yes but since I have the private keys I could move the coins elsewhere and claim my SV or ABC, right?

Did you moved your BCH to another wallet?

If you moved your coins to an exchange then you saved a lot of time for claiming both coins.