I happened to find an example in Huddleston and Pullum's The Cambridge Grammar of the English Language as an example of gapping:
(9) Tom will play the guitar and Mary sing. (p. 1542)
Thanks for sharing that example, Yasukotta. I hadn't checked to see whether CGEL dealt with the Gapping of a modal. I still question whether this is really Gapping and not simply a type of coordination that presents an illusion of ellipsis, though I also still need to run my idea by a specialist in this area.
In modern generative grammar, the subject of a clause is held to originate in the verb phrase—in the "Specifier" position of that phrase. It "raises" from that position to become the subject of the clause. Modals occupy a higher position in the syntactic hierarchy than the verb phrase.
Modals are also above Negation in the syntactic hierarchy, the NegP occupying a position between the modal position and the verb phrase. Now, consider that Huddleston and Pullum's example (9) can be made negative—provided "and" is changed to "or": "Tom will not play the guitar or Mary sing."
That sentence means the same thing as "Tom will not play the guitar, nor will Mary sing." But it is not an elided version of that sentence, and it is certainly not an elided version of the following: "Tom will not play the guitar, or Mary will (not) sing." That sentence does not have the same meaning at all.
These things being considered, it seems to me that a more appropriate analysis, even of such examples as (9), which does not involve negation, is that there is no ellipsis. Rather, there are two coordinated verb phrases beneath the modal, each with its own Specifier, and only the first Specifier raises. To illustrate: