O.K. I've read the whole thread and there sure is a lot of speculation from people who seem to have little to no knowledge of the effect an aftermarket downpipe has on a gen 1 MS3. I'll try to stick to facts and not speculate. It is a fact that aftermarket downpipes can cause the cam driven high pressure fuel pump to not be able to keep up. I use the word "can" because it does not follow that in her case this is the problem. When that pump cannot produce the ECU's targeted pressure, the way the car acts is much different than what she is describing.
I drove my '08 MS3 with my TurboXS catless dp and rp for over 30,000 miles with the stock pump and stock tune. I had no problems. In hindsight, I wish I had gotten a tune and upgraded pump internals before that, but for other reasons not germaine here. But, hear me out. Only when I added the Hypertech tune, which raises boost above stock levels, did I exceed the demand of the pump. And when that happens, you will know it. It won't just be sluggish or lose its pep. The engine will pull back timing to widen the spray of the injectors (longer duration) to try to compensate. When it can't, the low pressure event will be a momentary but pretty violent stutter, like load cut. I know, because that is what happened to me. The engine is trying to protect itself from the effects of a leaning out of the air to fuel ratio.
With respect, I think we can exclude the downpipe for this particular problem, although I agree that she will need to upgrade the pump internals just for peace of mind.
I think we can also exclude plugs as a cause, as the tell-tale sign would be engine misfiring. As far as I can tell, she is not reporting that. The car is just slow.
I agree with Bez that she should NOT put a tune on top of this mess until it is straightened out.
She needs something to monitor engine function. If she can't find someone with a dash hawk or other diagnostic equipment that can datalog, even a simple vacuum/boost gauge can tell a lot. I'm wondering if she can get someone to help her temporarily run a vacuum line into the cabin and attach a cheap $20 autoparts boost gauge. I would not be surprised to see that vacuum is normal (typically about 20-22 inches of mercury) but that boost is topping out at only about 10 psi. That is so-called "spring pressure." That can happen if there is a split vacuum line going from the turbo to the electronic boost control solenoid or from the solenoid back to the turbo inlet pipe. I chased a similar problem for quite some time and finally diagnosed a split vacuum hose.
She could also have a leaking bpv bleeding off boost. There could be a leak just about anywhere. With these kinds of leaks, the car will run and idle fine but never develop full boost at wide open throttle under load. The boost gauge would answer that question.
So she has an aftermarket downpipe. Is it catted or catless? What about the midpipe or race pipe? A damaged catalytic converter can produce constriction on the exhaust side. This backpressure can affect the ability of the turbo to produce maximum boost. We don't know what the previous owner did to the car. Checking for obstruction on the exhaust side needs to be done.
Another simple but possible answer would be an obstruction on the intake side, such as simply a dirty air filter.
What other mods, if any are there on this car? Another way is to use a stopwatch to time acceleration. A good and relatively safe option is for her to find a straight stretch of road and do a third gear wide open throttle pull from 50 to 75 mph. Published data for a stock gen 1 MS3 tells us that she should be covering that range in about 3.8 to 4.0 seconds, give or take a tenth or so. It it significantly slower than this, then something is wrong. If its acceleration is in that range, or quicker (given the downpipe), then she just encountered a quick Camry.