Publishing the same private app multiple times

Hello,

The situation is as follows: We have multiple clients that would be using the same app (possibly with different branding) and let their employees download it.

We are the owners of the app, the clients are just using it, so it has to be published under our account.

I understand it is not allowed to have the same or similar app published multiple times but does that also apply for private apps for businesses?

The reason we need this approach is because the backend of the app is deployed on-premise and different clients have different release lifecycles.

Thank you in advance!