common domain cookie question

Can someone clarify how this cookie is used?

 

If an SP is supposed to use the last-appended IdP doesn't this mean a principle is effectively authenticated to only one IdP at a time? 

 

1. user logs into IdP #1

2. user can access SPs honoring authentication via IdP #1

3. user logs into IdP #2

4. user can access SPs honoring authentication via IdP #2

 

5. user requests an SP honoring authentication via IdP #1 

6. said SP retrieves the common domain cookie, extracts last entry (IdP #2) and redirects browser to IdP #2

7. IdP #2 cannot authenticate the user for an SP requiring authentication via IdP #1 

8. what happens?

 

Is this correct?