Eh, it actually may ride a bit higher. The Si and LX stock heights are very very close. Also the Si springs are a higher rate, stiffer, to support the weight of the larger and heavier engine.
If you want a slight drop and a complete system the HFP kit is pretty awesome. Its struts and springs with OEM quality. It is for the SI but it does offer a slight drop from stock either way. We have an Si kit on my wifes LX. The drop is not very much at all but the improvement in ride and handling was noticeable. If the drop isnt enough definitely look into other aftermarket options like coilovers or drop springs.
Unlike the 8th gens, the 2012 Si and non-Si models use different front knuckles, so the Si suspension (HFP and non-HFP) doesn't fit on non-Si models. The Si uses larger diameter bolts to connect the struts to the knuckle and if I remember right, the hole spacing is different too.