I suppose however buys the "business" of Kodak film would certainly seek to use the Kodak name, and I don't see any difficulty in obtaining it. Kodak is already using its brand to market products of different technologies. By controlling the manufacture of the film they can also be certain that the Kodak name is not spent on something of low quality which would damage the other product lines.
The Del Monte brand was used by at least three different firms in different parts of the world until a few years ago (I don't know about now). I mean really three firms licensing the brand to produce their own products. If one had made some big mistake it would have damaged the market reputation of the other two.