There's nothing to claim, Satoshi can prove by signing a message, or moving his coins.
And that's what those claimers can't do.
A simple way to prove that they're satoshi.
But, they can't provide any signed message to prove that they're the owner of those bitcoins that satoshi has left, the real satoshi.
No one could provide any proof and evidence that they're the real Satoshi so no matter what they do, they couldn't support their claims. It's Satoshi's personal choice to keep his identity private so I wonder why these people are trying to pretend and steal his identity. He has chosen peace over anything else so he wouldn't show up instantly for no reason just like what fake Satoshis are doing.