No you do not change the W2 information you have. The IRS is looking to reconcile you tax return information with the same W2 information you have.
Once the recoup period is over, (note: it has been expanded to 31 Dec 2021), the employer will issue a corrected W2C to you with copies to the Social Security Administration to update their records. As you mention there is no change to your tax calculations only social security records. There is nothing you need to do.
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