Agreed that the opportunity for discussion and explanation should be provided to the employee. If false, fake, or fabricated information is given, don't recruit the employee.
Appointment Letter and Service Certificate
The labor laws, statutes, and enactments do mention the Appointment Letter and 'Service Certificate', with a prescribed form to be issued to all employees. The Service Card of employees has a column for commendations, but not for negative remarks.
Relieving Letter
A Relieving Letter signifies that the employee has settled all legitimate accounts, such as company property.
The comment in the Background Verification (BGV) is unclear as it does not mention the specific charge.
Notice Period and Employment Conditions
Assuming there was a notice period of 90 days for the employee and 30 days for the employer, which is arbitrary or discriminatory, or if intolerable conditions were created for the employee, or some vindictive or zealous line management or HR inserted comments to cause damage, then the next employer is falling prey to hearsay, gossip, or rumor.
The designated HR personnel could have issued and supplied the correct Full and Final (FnF) statement, adjusted notice pay, and demanded or paid the balance.
If a 90-day notice period is not applicable, then why should it be inserted in the appointment letter and enforced?
If the employee is aware and declines to serve the inapplicable notice period, then they shouldn't be penalized.
Background Verification Agency
Assuming that the BGV agency has not acquired information properly and can't prove it, then it has no right to be paid.
The employer should assess properly, and if the old company's Line Managers, HR, or third-party vendor that conducted BGV hasn't acted properly, blacklist them in internal records and appoint this employee. He will appreciate and deliver results for you.