SSH fails to ask for passphrase if a Zed update is available

Reproduction steps

1. Configure Git to use an SSH key with a passphrase.
2. Start Zed, with auto updates enabled.
3. Using the command palette, perform a Git operation that requires SSH (e.g. git: push or signed git: commit). Zed will ask for the passphrase (if not remembered by SSH agent) and complete the operation successfully.
4. Wait for an update to be available and the notification/restart prompt to appear in the title bar.
5. Again perform a Git action that requires SSH. Zed will not ask for a passphrase, and the operation fails with a generic error message. I think this happens even if a re-prompt shouldn't even be needed due to SSH agent.

I've seen this issue happen consistently for, I believe, every Zed update since I installed it earlier this year.

Current vs. Expected behavior

Current behaviour: When there's a pending Zed update available, Git operations that require an SSH passphrase fail without prompting for the passphrase.

Expected behaviour: Pending Zed updates should have no effect on whether Git operations and SSH passphrases work.

Zed version and system specs

Zed 1.26.2 Commit: c4273271092b0287f7d181f616bd5171e0a2f70c Version: 1.26.2+stable.384.c4273271092b0287f7d181f616bd5171e0a2f70c

Fedora 44 (KDE Plasma) Zed installed by install script (curl -f https://zed.dev/install.sh | sh)