$\hat{Z}$-invariants are q-series valued quantum invariants of (plumbed) 3-manifolds introduced by Gukov-Pei-Putrov-Vafa in 2017, and are known to give many examples of “spoiled” modular forms (e.g., mock/false theta functions). Gukov et.al expect the existence of logCFTs (non-rational VOAs) with $\hat{Z}$-invariants as their characters, but the mathematical study of logCFT is still underdeveloped and only a few easiest examples of such correspondences have been found. I have been studying that “easiest examples”, and in fact, it is not the VOA structure that plays a central role in that study, but rather a hidden purely Lie algebraic setting (shift system, arXiv:2409.07381). The point is that by using the geometric representation theory of Feigin-Tipunin construction developed in this setting (e.g., “BWB theorem”), we can qualitatively study the representation theory of that VOA without knowing the difficult VOA structures. On the other hand, assuming that the Weyl-type character formulas of the Feigin-Tipunin construction above can be applied recursively, surprisingly, we can obtain the $\hat{Z}$-invariants of Seifert 3-manifolds. In the recent preprint (arXiv:2501.12985), in rank 1 case, I defined an abelian category with a mechanism to derive the $\hat{Z}$-invariants in the above way in its Grothendieck group. This gives the (abelian) categorification of the $\hat{Z}$-invariants that has been expected by Gukov et.al, and also suggests a hypothetical but unified construction/research method for logCFTs, i.e., the theory of recursive shift systems (or nested Feigin-Tipunin construction).