When we isolate streams by domain, can a local observer guess how many domains we visit?

In Mike's future world where we allocate streams to different circuits based on their referer header (legacy/trac#3455 (moved)), that means the number of circuits we make will be a function of our browsing behavior (e.g. number of tabs we interact with).

How much can a local observer learn about the number of circuits we make based on seeing network traffic, and how much can that attacker deduce about our browsing activity based on number of circuits?

Do the answers change when we consider an entry guard instead?