Only show start project button in conversations

[mamoodi]

End-user friendly description of the problem this fixes or functionality that this introduces

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Only show start project button in conversations

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Give a summary of what the PR does, explaining any non-trivial design decisions

Tested that the Start new project button only shows when you are in the conversation page.

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Link of any specific issues this addresses
#5857

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To run this PR locally, use the following command:

docker run -it --rm   -p 3000:3000   -v /var/run/docker.sock:/var/run/docker.sock   --add-host host.docker.internal:host-gateway   -e SANDBOX_RUNTIME_CONTAINER_IMAGE=docker.all-hands.dev/all-hands-ai/runtime:58cb6b9-nikolaik   --name openhands-app-58cb6b9   docker.all-hands.dev/all-hands-ai/openhands:58cb6b9

[mamoodi]

@amanape thoughts on this solution? It's not that elegant so I'm happy to close if not desirable.
If it's fine, should we wait for Robert to confirm?