String you say. Well, I can believe that -- but it's still just a repellant force.
If it were weightless, but on a string, it would not stay tautly at the utmost end of the string, it would float around. Imagine a beachball floating in a pool, at the end of a rope -- but in three dimensions. The beachball does NOT strain against the rope at the far end of the pool - rather the rope just defines the furthest limit to which the ball can float. The same would be true for a genuinely weightless, but tethered, object.
A really telling experiment would be to create a very large such apparatus - like the size of a living room - and see if the levitated object would drift around, lazily bouncing off the walls, ceiling - and yes, the floor. If instead it flew to the ceiling and remained glued there, then we would know it had a strong upward force acting upon it, rather than merely a lack of downward force.
An even better, if far more expensive, experiment would be to put such an apparatus in orbit above the earth, then switch it on, and see if the object would start following a straight path away from the earth, no longer drawn by its gravity.
I *do* believe that some people are experimenting with "antigravity". I do not believe it is a proven science, let alone a mature principle ready for military applications.