OS X 12.3 breaks 1Password extension?

After upgrading my Intel MacBook Pro to OS X 12.3, I can no longer use the 1Password extension in any of my browsers.

In Firefox, the icon for the extension in the toolbar simply blinks on and off continuously. Clicking on it has no effect. Clicking the icon in the menubar produces a window telling me to install the extension.

In Safari, the icon in the toolbar does not blink, but clicking it has no effect. Clicking the icon in the menubar produces a window telling me to turn on the extension.

In Chrome, the icon is grayed out as if disabled. Clicking the window produces a pop-up menu whose first item reads "1Password extension (desktop app required).

The 1Password desktop app is installed, running and unlocked. I have removed and reinstalled the 1Password extension in Firefox and Chrome. I have upgraded 1Password from 7.9.2 to 7.9.3. None of these actions have resolved the issue.

Is there anything else that I should try? 1Password and its various extensions were working without problems before I upgraded the OS.

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1Password Version: 7.9.3
Extension Version: 4.7.5.90
OS Version: macOS 12.3
Referrer: forum-search:https://1password.community/search?Search=os%20x%2012.3