Motivated by chromosomes enclosed in nucleus and the recently discovered active topological glass, we study a spherically confined melt of long nonconcatenated active polymer rings. Without activity, the rings exhibit the same average large-scale conformational properties as chromatin fiber. Upon activating consecutive monomer segments on the rings, the system arrives at a glassy steady state due to activity-enhanced topological constraints. The latter generate coherent motions of the system, however the resulting large-scale structures are inconsistent with the fractal globule model. We observe microphase separation between active and passive segments without systematic trends in the positioning of active domains within the confining sphere. We find that tank-treading of active segments along the ring contour enhances active-passive phase separation in the state of active topological glass when both diffusional and conformational relaxation of the rings are significantly suppressed. Finally, although the present model of partly-active rings is not compatible with the large-scale chromatin organization, our results suggest that the activity-enhanced entanglements that result in facilitated intra- and inter-chromosomal contacts might be relevant for chromatin structure at smaller scales.