Adding a second server hangs indefinitely on "Connecting" with no errors in HA logs

Describe the bug

When adding a second existing Home Assistant server to the Companion App (while a first server is already configured and working), the app gets stuck on the "Connecting" state indefinitely. No login screen is ever shown. The server is discovered correctly, and entering the URL manually produces the same hang. The target server's own logs show no incoming connection attempt or error during the hang, suggesting the request is not reaching the server at all.

To Reproduce

Have one Home Assistant server already configured and working in the Companion App.
Go to Settings → Companion App → Servers → Add Server.
Enter the URL of a second, separate Home Assistant instance (discovered via mDNS or entered manually — both produce the same result).
Observe the app spins on "Connecting" indefinitely and never reaches the login page.

Expected behaviour

The app should connect to the second server and proceed to the login screen, as it does when this is the only/first server configured.

Additional context / troubleshooting already done

Confirmed the target server is reachable: logging in via the URL directly in Safari/Chrome on the same iPhone works without issue.
Confirmed it's not a Local Network permission issue (already enabled).
Confirmed no VPN, no reverse proxy, no SSL involved on the target server.
Confirmed it's not a network/discovery issue: tested adding this same server fresh on a different iOS device with no existing servers configured — connected successfully with no problems.
Target server's Settings > System > Logs show no errors and no record of the connection attempt during the hang.
Issue is reproducible specifically when a server is being added as a second server alongside an existing one.
Possibly related to #4681, which also reports broken behaviour specifically when a second server is added to the app — different symptom (location update hangs vs. initial connection hangs), but same general area (multi-server connection handling).

iOS device model, version and app version

Model Name: iPhone 13 mini

Software Version: iOS 26.5.1

App version: 2026.6.0 (2026.2018)

Home Assistant Core Version (target/second server): 2026.6.4

Start by reproducing the server-add flow in the Companion App with one existing server, comparing it with adding the same target on a device with no configured servers. Trace the second-server connection path and verify that the target server is contacted and the login screen appears instead of hanging on “Connecting”.

Written by the indexing model from the issue text.