I've tested it using the HTML file:

<a href="wpcom-local-dev://sync-connect-site?remoteSiteId=206672177&studioSiteId=2a1d3553-1ee1-4fe5-9297-4aeff02d9248">Connect</a>

1. Edit the file and fill out both site IDs
2. Open the file in the browser
3. Open Studio
4. Make the other site than the one defined in the test file active, open the Sync tab and then the 'Connect site' window
5. Click the Connect link in the test file

The PR doesn't fix this case. It seems that in this PR, we are changing the code to close the connection dialog immediately after the link is clicked. I'm unsure if it's the best approach, as the action the user just started is not yet finished, and keeping the dialog open would make more sense. User can change their mind, close the browser, and choose another site to connect in the connection dialog.

Should we update the sync-connect-site handle to close the dialog when the deep link is executed? And even further, to close any other dialog that could be open, e.g., 'Add site' or Settings?

@wojtekn, Thanks for testing the PR. This PR was made after exploring the other solution and it was an effort to find the simplest direct solution that solves user's pain.
I'll update this PR with the second approach, so the modal will still be open until the user finish the site connection.
I detailed both alternatives in STU-254 before implementing them, but in the end I thought to create the PR with the simple solution.

And even further, to close any other dialog that could be open, e.g., 'Add site' or Settings?

I see this a bit excessive at this moment. It would require a big refactor centralizing all the modals in the same Redux store. If you think it's necessary we can create a separate issue.

I see this a bit excessive at this moment. It would require a big refactor centralizing all the modals in the same Redux store. If you think it's necessary we can create a separate issue.

Agreed, I don't think we need that at this point.

This PR is ready for another review.