I thankfully haven't had to deal with MPI for a long time (and truthfully, my personal experiences have been pretty positive), but their website suggests they will have to pay the deductible up front, but should get reimbursed once the issue of who is at fault is sorted out.
Is MPI flat-out refusing to pursue this? If so, what exactly are they saying to justify it?
The answer might be to make themselves as irritating to the adjustors as possible...nothing to lose, really.
From the trailer, I don't get the impression that they're trying to come back from it - rather, they're leaning in to it.