Engine starts when it shouldn't??

My wife's 2012 Plug in Prius mysteriously starts the engine under these conditions: leaving home with full charge, she goes down a 650' long drive on a modest hill, maybe 15 -20 mph, touching the brake as needed to keep the car slow enough. Engine will fire up when the car hits about 12-15 mph. If she rides the brakes enough to keep the car below 9 or 10 mph, it won't start- upon reaching the bottom, she then accelerates modestly (maybe 35 mph tops) on the level and up another long, modest incline (all private lane, not public highway) and engine still doesn't start. Once on the road, EV works fine up to suburban highway speeds.
Only when the battery is discharged does the engine then cut in normally.

Not having her patience, I just put the car into neutral and let it roll, maybe to 25 mph, then slip into D when it hits the level, thereby avoiding starting the engine.

She claims this "starting" problem started just after we got new tires (same size as OEM, but different brand.

Seems like a programming issue, dealer was clueless.

 

Car is owned from new, 4 years or so. We live in Maryland, and the weather doesn't have to be cold for this to occur. We know that B will cause it to start, so we avoid that under these conditions. And yes, it begins with a full charge, after overnight. Seems to only happen on our drive, not otherwise... but then, it is her car, so I really don't know. As I had noted, she claims that this wasn't happening before we got new tires last spring, but I can't logically make a linkage.

I don't like driving it all that much. (Subaru Crosstrek and Mazda Miata are my rides)

 

Thanks, guys. I am starting to think the full charge, no place to put that new electricity might be the answer. I will ask her to brake with more force. What's confusing here is that if she keeps braking the car (and showing regen.) to keep the speed under about 9 or 10 mph, the engine doesn't start!

The tires are set at 3 lbs over what the manual states. Any more and the car bounces over bumps, although when it came home from the dealer, they had set them at the sidewall's max. pressure! Might help the mileage, but gives a ride that makes a track car feel mushy.

 

Ah, the "road" is a private lane, one car wide, going down thru the woods and over the stream. So going, say, 40 mph would be really pushing it! Like I said, doesn't start as long as she keeps it below 11-12 mph, but does if she goes 15 +-. Doesn't then start when she drives up the hill at normal speeds, maybe 25-30 mph in EV, or when she pulls out onto the main road and accelerates to 50 + mph in EV.
And no, I don't think it started on the hill for the first 3 or 4 years- same situation, otherwise. Tires are Michelin Energy Savers.