Is it true that an open restriction to $U \subset X$ induces a surjection on the set of irreducible perverse subquotients of perverse cohomologies (i.e. cohomologies with respect to the perverse t-structure) of an object of the bounded derived category of constructible sheaves? Is it true that this surjection sends to 0 exactly the subquotients of perverse cohomologies that are supported on $X-U$ and is a bijection from the set of irreducible subquotients of perverse cohomologies whose support intersects $U$ to the set of irreducible subquotients of perverse cohomologies of the open restriction?
Is it true that the Fourier transform (Fourier-Sato/Fourier-Deligne transform) induces a bijection on the set of irreducible perverse subquotients of perverse cohomologies of an object of the bounded derived category of constructible sheaves?