Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see his paper) corresponding to canonical basis vectors.

I'm interested in the stalks of these perverse sheaves, in particular, the stalk at 0. I believe I've reduced a (seemingly unrelated) conjecture to the question

Is the stalk of any one of Lusztig's perverse sheaves at 0 1-dimensional? (A weaker claim, which I think would make me roughly as happy is that the Euler characteristic of a stalk is 1.)

(The conjecture is that it is 1-dimensional, but I'm fairly agnostic on this point. I wouldn't be surprised either way, though if I had to choose I'd say this conjecture sounds a little unlikely).

Is there a proof or counterexample of this claim in the literature?

After a quick read-through of some the literature (Lusztig's paper, Reineke's paper on resolutions, Kashiwara and Saito's paper on crystal structures on components, etc.) I'm not feeling any closer to understanding these stalks, and am having trouble finding anywhere else to look.

**EDIT**: This seems like an even longer shot, but what would be even better would be to understand the cohomology of one of these sheaves on a "quiver orbital variety,'' the space of quiver representations (not modulo isomorphism!) which preserve a particular flag. Is there anything about this in the literature?