Insufficiency of the algebraic Brauer--Manin obstruction for homogeneous spaces

Over any number field containing a root of unity of odd prime order, we construct a homogeneous space of ${\mathrm{S}\mathrm{L}}_n$ with finite $2$-nilpotent geometric stabilizers, with a constant unramified algebraic Brauer group, which has no rational point but has local points in every completion of the ground field. This yields the first example of transcendental Brauer–Manin obstruction for homogeneous spaces of connected linear algebraic groups. Our method exploits a previous idea by Borovoi and Kunyavskii.