  We prove that the moduli spaces of twisted $\mathrm{SL}_n$ and
$\mathrm{PGL}_n$-Higgs bundles on a smooth projective curve have the same
(stringy) class in the Grothendieck ring of rational Chow motives. On the level
of Hodge numbers this was conjectured by Hausel and Thaddeus, and recently
proven by Groechenig, Ziegler and the second author. To adapt their argument,
which relies on p-adic integration, we use a version of motivic integration
with values in rational Chow motives and the geometry of N\'eron models to
evaluate such integrals on Hitchin fibers.
