Fukaya's immersed Lagrangian Floer theory and microlocalization of Fukaya category

Let 𝔉𝔲𝔨(T^*M) be the Fukaya category in the Fukaya's immersed Lagrangian Floer theory which is generated by immersed Lagrangian submanifolds with clean self-intersections. This category is monoidal in that the product of two such immersed Lagrangian submanifolds remains to be a Lagrangian immersion with clean self-intersection. Utilizing this monoidality of Fukaya's immersed Lagrangian Floer theory, we prove the following generation result in this Fukaya category of the cotangent bundle, which is the counterpart of Nadler's generation result for the Fukaya category generated by the exact embedded Lagrangian branes. More specifically we prove that for a given triangulation 𝒯 = {τ_𝔞} fine enough the Yoneda module 𝒴_𝕃: = hom_𝔉𝔲𝔨(T^*M)(·, 𝕃) can be expressed as a twisted complex with terms hom_𝔉𝔲𝔨(T^*M)(α_M(·), L_τ_𝔞*) for any curvature-free (aka tatologically unostructed) object 𝕃. Using this, we also extend Nadler's equivalence theorem between the dg category Sh_c(M) of constructible sheaves on M and the triangulated envelope of 𝔉𝔲𝔨^0(T^*M) to the one over the Novikov field 𝕂.