Boundary determination of odd-dimensional handlebodies

Prove that any two (2k+1)-dimensional k-handlebodies with diffeomorphic boundaries are themselves diffeomorphic.

Background

The paper conjectures that a (2k+1)-dimensional k-handlebody is determined up to diffeomorphism by the diffeomorphism type of its boundary. The conjecture is immediate for k=0 and k=1. For ke3, Lawson's result gives only stable diffeomorphism: after taking boundary connected sums with sufficiently many copies of Sk44 B{k+1}, the two handlebodies become diffeomorphic. Thus, the unresolved issue is whether stabilization can be omitted.

The conjecture would imply that in a decomposition of a closed (2k+1)-manifold into two k-handlebodies with common boundary, the two pieces are necessarily diffeomorphic. The paper notes that this implication would yield Lawson's theorem on decomposing closed orientable (2k+1)-manifolds into two diffeomorphic k-handlebodies.

References

Let $X$ and $X'$ be $(2k+1)$-dimensional $k$-handlebodies. If their boundaries $\partial X$ and $\partial X'$ are diffeomorphic, then $X$ and $X'$ are diffeomorphic.

— A note on some high-dimensional handlebodies  (2502.11314 - Kim, 16 Feb 2025) in Conjecture labeled (conj), Introduction

It remains unclear whether $X_1$ and $X_2$ are necessarily diffeomorphic.

— A note on some high-dimensional handlebodies  (2502.11314 - Kim, 16 Feb 2025) in Remark immediately following Conjecture (conj), Introduction