  For the group $\mathrm{GL}_{d}$, we confirm a conjecture of Goresky, Kottwitz
and MacPherson, which states that the cohomology of the affine Springer fibers
depend only on the root valuation datum of their defining elements. The proof
relies on a microlocal analysis of the intermediate extensions appearing in
Ng\^o's support theorem, and a Whitney regularity property of the union of the
strict $\delta$-strata over the equisingular strata for the miniversal
deformation of plane curve singularities.
