seems like they own that wallet with 1000+ ethereum based on my personal research.
Your research is based on the ~1300 ETH address sending some amount to the 10 ETH address.
This is not a good way of establishing proof of ownership since I can do the same thing by withdrawing Ethereum from an exchange. The exchange's input address would most likely have a huge amount of Ethereum, right?
Simplest way is to sign a message.