Attaching a smooth handle with corner rounding

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $X$ be a smooth $n$-manifold with boundary, and let $k$ be an integer with $0\le k\le n$. Attach the handle of K handle core cocore attaching region and belt sphere by a smooth embedding $h:S^{k-1}\times D^{n-k}\to \mathrm{\partial }X$ that extends to a neighborhood of the disk factor. Form the quotient of $X\bigsqcup (D^k\times D^{n-k})$ identifying $z$ with $h(z)$ in the attaching region. The disk coordinates trivialize the normal bundle of the attaching sphere; this framing is part of the data. Use collars from Collar neighborhood theorem to give the seam its product smooth charts, then round the compact codimension-two corner. A compatible rounding is a smooth monotone planar profile, transverse to a common diagonal direction, agreeing with the two faces away from a small corner neighborhood. In coordinates along that diagonal it is a graph. This convention fixes the gluing and collar data; changing the attaching embedding is a different question. There is no corner to round when $k=0$ or $k=n$.