Application management within the Private Space is limited to only company-owned work profile devices. Personally owned devices - or BYOD - cannot manage the Personal Space in any way. Fully managed devices do not support Private Space.
For company-owned work profile devices enrolled into AMAPI, the personal usage policies of personal Play Store mode and personal applications apply both to the parent (personal) profile, and the Private Space.
If your EMM uses a custom DPC with Play EMM API, this feature will be available in future. Until then it is not possible to restrict applications within the Private Space.
via, updated by Google on 2024-11-12 to clarify custom DPC status.
If your organisation already leverages policies to allow or block applications in the parent profile on a COPE device, these will carry over automatically to the Personal Space.
Note: It is not possible to set different personal Play Store modes, or configure separate allow/blocklist policies between the parent profile and the Personal Space.