database design – How to model this scenario conceptually? Bridge entity needed?

I have a roommate entity(primary key: identifier of the shared house) and housekeeper entity (primary key: housekeeper's identifier).

At any time, a housekeeper can only work in one shared dwelling.

When a housekeeper finishes an assignment with a particular house share, the allocation is recorded (start date and end date).

A member of the household staff may return to a colocation in which he previously worked; if they do, a new allowance is recorded.

I believe the relationship between the house and the housekeeper must be multiple. I can come with bridge entity (for example, housework) who have Housekeeper ID, start_date and end_date of pk. But generally, a bridge entity will have primary keys of both entities but not in this case. I wonder if I model what's wrong?