Everone is kinda right. On factory air boxes there is a bunch of design critera. Two of them are a resonant chamber to aid in airflow to the engine and resonance to cut noise. The resonance aspect of the irbox does not really change with boost so a properly designed large boost box will help airflow at a given rpm increasing power. However I doubt that any of the turbo companies are making air boxes with that in mind. Most are doing what they can for packaging into the tight spaces. So having a larger air box will give a cushion to the boost pressure as the turbo pressurizes the airbox and will also take longer to fall off when the throttle is realesed. A BOV helps this alot. The problem with the silicon tube style is that it is an insulating material and does not have the surface area to expel heat that the air box would have. Having injectors in the chargetube/airbox/throttle bodies only helps the charge temp a small amount from the difference in temp from the fuel to the air temp in the box, because it is a pressurized injector there is no heat loss from latent heat of vaporization that carbs benifit from.
For hp the best will always be a oversized intercooler that is effecient. the water to air designs are the best but do make for a very long charge side of the turbo sytem. This does effect response.