Presentation conjecture for the non-compact 3-dimensional cobordism 2-category

The objects are closed 1-dimensional manifolds equipped with chosen bounding 2-manifolds, the 1-morphisms are 2-dimensional cobordisms equipped with bounding 3-manifolds, and the 2-morphisms are 3-dimensional cobordisms with corners equipped with bounding 4-manifolds, as described above. Let $\mathbf{Bord}^{\mathrm{sig},\mathrm{nc}}_{3,2,1}$ denote this non-compact 3-dimensional cobordism 2-category. Presentation conjecture. The non-compact 3-dimensional cobordism 2-category $\mathbf{Bord}^{\mathrm{sig},\mathrm{nc}}_{3,2,1}$ is equivalent to the 2-category obtained from the presentation described below. This would give a finite generators-and-relations description of the source 2-category for the associated TQFT; the equivalence is presented as conjectural here, and no resolution is supplied in the source.