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Let $V,V'\subset X$ be smooth subvarieties of a smooth projective variety. Denoting $i_V, i_{V'}$ the inclusion, is it true that if $i_{V,*}c_i(T_V)=i_{V',*}c_i(T_{V'})$ in $\mathrm{CH}^*(X)$ for all $i$ then $i_{V,*}(c(T_V)^{-1})=i_{V',*}(c(T_{V'})^{-1})$ in $\mathrm{CH}^*(X)$?
It seems to be wrong since the inverse uses intersection theory on $V$ (resp. $V'$) but I do not have a counterexample?

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