Existence of adapted metrics for sectional-hyperbolic sets

A sectional-hyperbolic set is an invariant set with a partially hyperbolic splitting whose contracted bundle is uniformly contracted and whose complementary bundle is sectional-expanding. Adapted-metric conjecture. There exists an adapted metric for all sectional-hyperbolic sets for any $C^1$ vector field in any dimension. This extends the adapted-metric construction proved in the paper's three-dimensional setting to arbitrary dimensions and is presented as an open direction concerning more general forms of sectional expansion.