chore: Enforce Do Not Merge check into sync settings

[dandhlee]

Checklist

• [x] I have followed Sample Guidelines from AUTHORING_GUIDE.MD
• [ ] README is updated to include all relevant information
• [ ] Tests pass: nox -s py-3.9 (see Test Environment Setup)
• [ ] Lint pass: nox -s lint (see Test Environment Setup)
• [ ] These samples need a new API enabled in testing projects to pass (let us know which ones)
• [ ] These samples need a new/updated env vars in testing projects set to pass (let us know which ones)
• [x] Please merge this PR for me once it is approved.
• [ ] This sample adds a new sample directory, and I updated the CODEOWNERS file with the codeowners for this sample

[kweinmeister]

@dandhlee is this still something you plan to get around to?

[dandhlee]

@kweinmeister I haven't had the chance to look into this. I might be able to get to it but won't have time in the foreseeable future as there were multiple moving parts that I needed to look into before getting this working :/

[iennae]

As this PR is over a year old now I'm going to close it. If you want to reopen and address the concerns raised, please feel free to do so. Thank you!