  We prove that the (twisted orbifold) motives of the moduli spaces of
$\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles of coprime rank and degree
on a smooth projective curve over an algebraically closed field in which the
rank is invertible are isomorphic in Voevodsky's triangulated category of
motives. The equality of twisted orbifold Hodge numbers of these moduli spaces
was conjectured by Hausel and Thaddeus and recently proven by Groechenig,
Ziegler and Wyss via $p$-adic integration and then by Maulik and Shen using the
decomposition theorem, an analysis of the supports of $D$-twisted Hitchin
fibrations and vanishing cycles. Our proof in characteristic zero combines the
geometric ideas of Maulik and Shen with the conservativity of the Betti
realisation on abelian motives; to apply the latter, we prove that the relevant
motives are abelian. In particular, we prove that the motive of the
$\mathrm{SL}_n$-Higgs moduli space is abelian, building on our previous work in
the $\mathrm{GL}_n$-case. We then use motivic nearby cycles to deduce the
result in positive characteristic from that in characteristic zero. Using the
same ideas, we prove motivic $\chi$-independence for $\mathrm{GL}_n$-Higgs
bundles.
