A friend, Alice, applied to a few jobs. Some didn't pan out, then the best fitting one offered a job. Alice accepted that job verbally, received a background check clearance form and signed it, and now they're waiting the ~7 days necessary for the background check to complete and an official job offer to be made and accepted.
During this time Alice is wary of taking any action based on the job offer. For example, Alice's current position asks for 2-4 weeks notice and the new employer encouraged her to give notice to her current job, assuring that the background check is just busy work and they have no doubt Alice will pass. It is likely Alice will pass, no expected issues with her background or eligibility for the job. Alice is going to wait and only give notice once a formal job offer is signed by both parties.
The question is about an interview Alice has scheduled for another job. This is a decent fitting job, though not as good as what she got offered and (unofficially/pre-background check) accepted. On the one hand, it's risky and rude to continue interviewing for other positions after having accepted a job. On the other hand, Alice feels she hasn't 100% gotten a new job, so it feels risky to withdraw from other opportunities. This alternate job is a 2nd/3rd round interview so she is far into the process, but she expects she'd have a full job offer before she has to make a decision at this alternate job (and if forced to decide, she'd say no to anyone other than the tentatively-accepted job of course).
What is safest/most professional in this case? Proceeding with the final interview round at the alternate job, or withdrawing from it ASAP? There's probably 3-5 days between interview at alt job & completion of background check at tentative new job. It's important to note that the two potential employers are separate companies but closely related work, and it wouldn't be surprising if management from both places interacts socially.