Let (X,τ) be a topological space and let τh(X) denote the topology formed by its h-open subsets. We study the separation axiom h-R⋆ through an intrinsic formulation in the associated space (X,τh(X)), using cover-boundedness in its trace subspaces. Relative cover-boundedness is shown to imply global h-boundedness. We prove the strict hierarchy h-R1 ⇒ h-R⋆ ⇒ h-R0 and give explicit examples showing that neither implication reverses. The examples arise from two natural h-fixed topologies obtained by isolating one point in cofinite- and cocountable-type spaces. We also show that strong h-regularity implies h-R⋆, characterize h-submaximality through the associated topology, and provide an explicit h-submaximal example. A classical heredity result yields heredity for h-subspace compatible subspaces. Under h-product compatibility, arbitrary products of h-R1 spaces and arbitrary products of strongly h-regular spaces are h-R⋆. Finally, we establish invariance under h-homeomorphisms and introduce a ≪h-lifting condition that gives a surjective preservation theorem. Perfect maps between the associated spaces automatically have this lifting property, and a compact inverse image hypothesis provides a further sufficient criterion.