The 70-200/4G OSS already shows that full format mirrorless cameras make rather limited sense.
Putting a 300/2.8 on such a camera would be downright ridiculous IMHO.
Sony should look into Canon's DO technology maybe - despite some problems this is (so far) the only thing that make sense for long tele lenses on mirrorless cameras.
This will be an APS-C lens of unknown speed at least.
400mm f/5.6 lens still requires at least 77mm front filter, no matter what format it is. Do you think it's going to be slower, like f/6.7?
Personally I'm expecting a 4.5-6.3 lens. But it'll not be small - Fuji isn't into small zoom lenses.