Can you prevent Snowsight from using Account URL when using client redirect/connection url?

My snowflake org is "foo"

I use a "connection object" named "bi", so my connection url is "foo-bi.snowflakecomputing.com". That is what I want the company's users/business partners to see.

That url allows logging in to a Snowflake connecting to an account "foo-biprod" in Snowsight, snowflake for python connector, and other software that needs to query objects in the account.

After logging into Snowsight, the "ugly"/"confusing" account name is visible in the browser's address bar, and some of my team object to the account name being visible.

  1. Is there any way to prevent Snowsight showing the account name when you use a connection URL to sign on?
  2. Does any user-facing client software make the account name visible? I can't think of anything myself -- business users use a SQL client or a BI tool. They configure it with the connection url.
  3. If other accounts create shares to us, they do have to have our consumer account name(s) that can/might be reading from Share? I think it is not possible to share to a connection URL, and publisher would need account name(s). From example in docs I am not sure if org-acct syntax is supported anyway in shares. (The example is ALTER SHARE sales_s ADD ACCOUNTS=xy12345, yz23456;).
  4. If we create shares to other accounts, is the name of our account needed by the owner of the consumer account?

Any other places/audiences that the account name is going to leak and we can't use the connection url? E.g. SaaS vendors?