Existence of non-trivial Fourier–Mukai pairs among triangularizable spaces

Determine whether there exist non-homeomorphic locally compact triangularizable spaces X and Y whose derived categories of sheaves of abelian groups are equivalent, namely whether mathrm{Shv}(X)[mathrm{D}(mathbb{Z})]simeq mathrm{Shv}(Y)[mathrm{D}(mathbb{Z})].

Background

The paper defines a non-trivial Fourier–Mukai pair as a pair of non-homeomorphic topological spaces with equivalent derived infty-categories of sheaves of abelian groups. It proves that no such pair exists when the first space is second-countable, triangularizable, and either one-dimensional, a finite-type surface, or the 3-sphere.

The authors note that non-trivial examples arise from non-sober or Alexandrov spaces, which are regarded as point-set-topological pathologies, but report no example among triangularizable spaces. Thus the unresolved issue is whether non-trivial Fourier–Mukai partners exist in the broader class of “nice” triangularizable spaces outside the cases settled by the reconstruction theorem.