How does Coordination of Benefits (COB) work in practice?

Context: I currently have a COBRA policy under which I surpassed the out-of-pocket limit. I will start a new job next week and my new health coverage will kick in straightaway. Further details here.

The COB verbiage in both plan documents (current and new) states the following:

If it is determined that this Plan is the secondary payer, benefits will be adjusted and reduced (standard). Benefits payable from both plans shall not exceed 100% of the eligible U&C charges.

Question: If I keep my current COBRA policy to pay for a scheduled expensive procedure next month, how will the primary and secondary coverage work in practice? Let’s consider the mock example below.

Example:

Patient goes to hospital for surgery and stays overnight for the recovery.

The total cost for the procedure is billed as $100,000.

The new (primary) insurance determines that the individual is liable for the entire out-of-pocket limit of $10,000 and the insurer covers the remaining $90,000.

Will the older (secondary) insurance, for which the out-of-pocket limit had already been spent, cover the above mentioned $10,000? In other words, is the individual made whole despite the potentially cumbersome bureaucracy, or is he or she going to be out of $10,000?