Yes, but gravity where? That is the question.
Gravity "associated with" an object, does not necessarily exert an influence on that object - that is the point.
You are saying "no, there is energy stored elsewhere, so the transference of gravity is put off by that energy"... when - directly speaking - it does not.
Once again, what you are saying does not fit with what science has told us about gravity.
Every object with mass creates gravity. Mass creates curves in spacetime, just the same way that a bowling ball on a trampoline will cause a curve in the trampoline's surface. This curving is gravity. And I can roll a marble across the trampoline, and when it passes close to the bowling ball, the curve cause by the bowling ball will change the direction of the marble. Likewise, if a small object like a space probe passes close to a large object like a planet, then the curve in space time caused by the planet - the planet's gravity - will make the space probe change direction.
This curve of space time technically goes on forever. If there was a whole universe with just you and a single planet, then no matter how far away you go from that planet, you'll still feel its gravity. Of course, the strength of the effect you'd feel would fall off very quickly. It's like how a light bulb can be blindingly bright if you are right next to it, but if you are in the next room, you can look at it without hurting your eyes. Likewise, you can get so far away from this one and only planet in our hypothetical universe that you can't consciously feel the effects of its gravity. But the effect is still technically there, just incredibly small.
In the real universe, the gravitational effects of distant objects like stars and planets is very small, and it's drowned out by the gravitational effects of the moon and the sun. But that gravitational influence is there nonetheless.
So, keeping this in mind, what exactly do you mean by "Gravity "associated with" an object"?