Just adding a reference to the literature where this question seems to have cropped up :
"Calculating canonical distinguished involutions in the affine Weyl groups" - Chmutova, Ostrik (pdf)
They phrase the question as : "Are all Richardson orbits strongly Richardson ?" ('strongly Richardson' being their term for the existence of a resolution satisfying the necessary properties). The published version of the paper (pdf link above) also includes a family of counter examples(== one of the examples alluded to in Ben's answer).