Truncated factorized perverse sheaves on Sym(C)

Kapranov and Schechtman defined the category FP of factorized perverse sheaves on Sym(C) smooth along the stratification given by multiplicities and with values in a braided monoidal category V. We define for each d in N the category FP^{\leq d} of factorized perverse sheaves on U_{n\leq d}Sym^{n}(C) and the category FP_{\leq d} of factorized perverse sheaves on the open subset of Sym(C) consisting of multi-sets with multiplicities bounded by d.
We prove that the natural restriction functor from FP_{\leq d} to \FP^{\leq d} is an equivalence for any d in N, and that FP^{\leq 1} and \FP_{\leq 1} are equivalent to V. We show that the full direct image *, the extension by zero ! and the intermediate extension !* induce functors from FP_{\leq d} to \FP.
In addition, we show that the families (FP^{\leq d})_{d in N} and (\FP_{\leq d})_{d in N} fit into systems of categories, compatible with restrictions and extensions, whose inverse limit is FP.