One theory that is being floated on Reddit runs like this:
Kleiman is Satoshi, and had the keys to the ~1 million bitcoins. He dies, and his USB stick/computer/whatever went to a relative, who doesn't realize what he is holding. Wright knew Kleiman and knew he was Satoshi. So he invents this crazy story about being Satoshi, but that he can't spend the coins because they are all in a trust that was held by Kleiman.
So now Wright comes public claiming to be Satoshi - and sets himself up to launch a lawsuit against Kleiman's relative to get "his" bitcoins back. If Wright pulls this off, he gains the fabled treasure of 1 million bitcoins off Kleiman's estate.
Thoughts pro and con?
Wright might be Satoshi, but not the programmer or cryptographer, but the spokesman. Kleiman was doing the coding and crypto staff, and Wrigth was his mouthpeace. Wright was posting about bitcoin, contacting others by email (Back, Wei, Finlley), writing/helping with the white paper, organizing servers and forums for bitcoin etc.
So technicaly. Wright could be the person writhing under pseduonim Satoshi nakamoto. It does not mean he was alone, nor that he was coding Bitcoin and crypto stuff.