Background Check Dates

Hey everyone, hope you are all doing well. I will be interning in IB this summer and had a question about background checks. I completed the background check about 4 weeks ago. For one of my internships, I listed the dates as September 2019 -- December 1, 2019. However, my ex-boss verified the internship dates on the background check as September 2019 -- November, 4 2019. Looking back at the emails, I actually did work up until the beginning of December. I emailed my ex-boss after the background check was completed and he said that he did not remember the exact dates. My submitted resume also had the dates as September 2019 -- December 2019. There is almost a full month between the end date I provided versus the end date my ex-boss provided.

Since it has been several weeks since the background check, should I still reach out to HR? Or is it a better idea to wait for them to contact me if they deem this discrepancy as important? I've seen some people advise reaching out to HR while others advise to wait for them to reach out. Just wanted to know what to do so I know what to do in the future. Thanks you and stay safe.