Wensleydale. Sorry you are wrong.
The E-3s and E-6s are both derived from the Boeing 707-320 model, hence they come from the same airframe model. They are NOT different. It would make NO difference if an E-3D had an E-6A wing substituted during assembly since they are both powered by the F108 (CFM 56), hence the same design and attachment points.
Not sure where you "heard" this story, but, if true, is just an interesting bit of trivia. Has absolutely NOTHING to do with safety, airworthiness or operations of the E-3Ds.
BTW, the original Dash 80 (132in diameter) was modified a bit to develop the KC-135 (144in) and modified with a wider fuselage (148in for six-abreast seating), larger horizontal tail surfaces and modified wings, as explained before.
I suggest you read the excellent book on the 707/135 by Dominique Breffort: 707, Boeing KC-135 and Their Derivatives, by Historie & Collections. Might provide some actual facts to go along with 'rumors'.