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You are right, I made a mistake (I momentarily forgot that the Euler class of a non-orientable bundle lives in twisted cohomology...). So $w_2 + w_1^2$ must be the mod 2 reduction of a class in twisted cohomology. Then I think that $X =\mathbb{RP}^4$ provides a counter-example. If I did my back of the envelope calculations right (please check) then $w(TX) = 1 + x + x^4$ and so $w_2 + w_1^2 = x^2$ is non-trivial, but I think $H^2(X, \mathbb{Z}^{w_1}) = 0$, so this class can't be the reduction of a twisted cohomology class.
I am guessing that you mean that your structure is the same as a spin structure on $TX \oplus \mathcal{V}$ where $\mathcal{V}$ is a real 2-plane bundle. The obstruction for this is the existence of an integral lift of $w_2 + w_1^2$ (is this your "twisted integral lift" of $w_2$?). Doesn't $\mathbb{RP}^2 \times \mathbb{RP}^2$ give a counter-example?
@PavelSafronov In D.1.3 of Appendix D in Gaitsgory's paper he says "... it is easy to show that O is rigid in the sense of Sect. D.1.1 if and only if every compact object of O admits both left and right monoidal duals". However right now I am only able to see that this implies "weak rigidity" as in Bakalov-Kirillov's Lectures on Tensor Cateogries and Modular Functors. Do you understand the argument here?
I believe these structures only give "weak rigidity" and not rigidity itself. See for example Bakalov-Kirillov "Lectures on tensor categories and modular functors".
Then from this we see that the class of what Clark and I call the "gaunt" categories is preserved since those are the categories such that $hoCat(G,C) = hoCat(pt,C)$ for all groups G. Again we have hoGaunt = Gaunt, has only two automorphisms and we may assume WLOG that it is the identity automorphism on Gaunt. Using this I think I can show that for any category C we have that C and F(C) have equivalent presentations (free categories on directed graphs are gaunt). So I think any such functor must be the identity on objects of hoCat. It could a priori still be interesting on morphisms?
Here are some thoughts: Since the equivalence class of the terminal category is preserved, we can show that the collection of categories equivalent to posets is preserved. (They are the categories P such that $hoCat(D,P) \to Set( hoCat(pt, D), hoCat(pt, P))$ is injective for all cats D). Now hoPoset = Poset only has $\mathbb{Z}/2$ many automorphisms. WLOG we can assume our functor is the identity on Poset. So then the free-walking arrow is preserved, identically. From this it follows that the class of groupoids is preserved (as well as the connected groupoids which I will call "groups"). cont
