Uniform non-integrability for singular-hyperbolic attracting sets

Let an ergodic physical probability measure be supported on a non-trivial singular-hyperbolic attracting set. Uniform non-integrability conjecture. The Uniform Non-Integrability (UNI) condition holds for all ergodic physical probability measures supported on each non-trivial singular-hyperbolic attracting set. UNI is known in the Lorenz-like case with a unique Lorenz-like singularity and an ergodic physical probability measure; the conjecture asks for the corresponding result for general singular-hyperbolic attracting sets with finitely many singularities, which remains open.