There used to be many candidates for an exotic 4-sphere, but a lot of them are now known to be the standard smooth $S^4$. The ones of Cappell-Shaneson (maybe not all of them?) were described in terms of handlebody decompositions where there are no 3-handles, but I think the proofs that these were diffeomorphic to $S^4$ involved altering the decompositions with a cancellation trick by introducing some 3-handles.
I am not sure what is left over, but I seek the following:
1) Is there a potentially exotic 4-sphere which is prescribed by an explicit 4-handlebody decomposition with only 0,1,2-handles and a single 4-handle?
2) Of the standard Cappell-Shanelson spheres with a given handlebody decomposition involving no 3-handles, is it known that we can prove [edit: some of them] they're standard without the trick involving 3-handles?
The reason I ask is because I'd like to study certain differential 2-forms on these spheres which "play well" with the handles, and to understand the deformation of these 2-forms as we change the handlebody decompositions.