In https://arxiv.org/abs/0807.2431, Honda--Kazez--Matic introduce a definition of (1+1 dimensional) sutured TQFT; see also e.g. Mathews http://arxiv.org/abs/1006.5433 and Fink https://arxiv.org/abs/1210.0238. This construction morally comes from symplectic / contact topology and has many relationships with contact categories, bordered sutured Floer homology, etc.

On the other hand, bordered (or bordered sutured) Floer homology can be extended down one dimension to ``cornered'' Floer homology, by Douglas--Manolescu and Douglas--Lipshitz--Manolescu. Their theory uses nilCoxeter algebras with differentials, which resemble KLR algebras or 2-morphism spaces in categorified quantum groups. In fact, one can show that interesting relationships exist between cornered Floer homology and the 2-representation theory of $\mathcal{U}_q(\mathfrak{gl}(1|1))$.

This type of representation theory, especially at the decategorified level, is usually related to TQFTs via the Reshetikhin--Turaev construction or other similar methods. The Reshetikhin--Turaev TQFT and its cousins, beginning with a modular category $\mathcal{C}$, don't seem "sutured" in any sense. They're either extended $1+1+1$ TQFTs satisfying all the axioms if $\mathcal{C}$ satisfies the usual modular category axioms, or (using constructions of Kerler and Lyubashenko) they're extended "connected TQFTs" or (even better) "extended half-projective TQFTs" if $\mathcal{C}$ satisfies similar but weaker axioms which don't imply semisimplicity. But suture-like structures aren't part of the story.

My question is: from any "abstract TQFT perspective" (e.g. modular categories and Reshetikhin--Turaev, the cobordism hypothesis, or some other perspective), is there a way to fit sutured manifolds into the picture? The cobordism hypothesis has various versions where the manifolds involved are decorated somehow; is there a way to view a sutured TQFT as a more ordinary TQFT with some type of decorations? And is there any explanation, starting abstractly rather than with symplectic or contact manifolds, why sutured manifolds and their invariants might arise in a TQFT setting?