Help with a desk build

The idea so far is a 6 1/2ftx2 1/2ft desk. Made with 3/4" hardwood plywood (haven't decided which wood), 1x3 hardwood edge strips (most likely going to be oak), and the legs being welded from 2" OD tubing with a 4" flange screwed into 5/8" threaded inserts. I'm concerned that it wont be stable laterally with the 4 independent legs as well as sag towards the center of the desk. I have thought about adding a 5th leg centered on the back edge but am not sure if it is necessary.

Here's a SketchUp model I put together: Imgur