or are you connecting the FE to the cloud BE?

After opening the conversation on the desktop app, each one start showing up on the web app

so there is some sort of state issue there

When you open a conversation in Canvas, it wakes up the cloud sandbox if it was paused, and the cloud only reports an agent status for conversations whose sandbox is running. The "Active only" thread filter in the sidebar uses exactly that status. So if the browser tab is on "Active only" (the Active and Compact views both turn it on), you'd only see conversations with a running sandbox, which is your two, and each one would appear as soon as the desktop app wakes it up. Could you check in the browser tab: the View picker above the conversation list, and More options → Threads? If it says Active only, switching to All threads (or the Recent view) should bring everything back. If it already says All threads, please send me the org shown in the org switcher in both apps and the desktop app version, and I'll dig further.