Do you have a preference for the second job or is this maximizing earnings? If you much prefer the second then taking the risk and letting the process play out with the second might be worth it. One option with the second job is once you have the first offer in hand just tell them that you received another offer but would prefer there position if they can make a final decision by X date.
I know when I have done first rounds of interviews and then there is a manager/exec interview that not once has it changed the guy I wanted. So as long as you have had an in person with your future boss they have enough information to make a decision and if your future boss wants you that much they will make it happen not to lose you. If they aren’t accommodating about getting you an offer within a week you probably aren’t their first choice.
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