Put simply, you’re doing them a massive favour by providing tires, so I see no reason why they wouldn’t. If the won’t, then have their maintenance plan tires, pay for changing yourself and sell there’s with a few miles on to cover your costs??
My story might be helpful… my GTD was on LEX autolease and included maintenance plan, when the time came for new tires I went to kwikfit as per the agreement who replaced the mediocre Bridgestones with some eco friendly, soft side walled absolute crap. I thought they’d broken the car as the handling was almost completely gone.
To say the least, I was not impressed, they refused to change them no matter how much I complained. My company didn’t care so wouldn’t help me contact LEX, in the end, I just burnt through the tires as quickly as possible, deliberately wheel spinning at every safe opportunity.
When it was time again I kicked up such a stink at kwikfit and almost fell out with them until I got to speak to the manager. I insisted they put on the correct OEM spec tire, at the end of the day LEX charge us maintenance plans based on the spec of the car, not generic prices. Eventually they agreed and fitted the latest Continental sport contact tires which were pretty decent.
The moral of this is, whilst I didn’t get UUHP like the PS4s, if your car came with Goodyear eagle supersports (like my CS45) you’re well within your rights to insist they fit these or equivalent tires.