The burn-in problem was caused by the drastic reduction of the pixel size. The resolution of the 7.9-inch iPad mini with Retina display is 2048×1536, about four times clearer than the existing 1024×768 products. LG Display used the amorphous silicon (a-Si) for the substrate, whereas Sharp used IGZO. The pixel of the smartphone display is smaller than that of the iPad mini Retina panel, but as the Low Temperature Polycrystalline Silicon (LTPS) technology is used, it can be produced stably. The pixel of the iPad mini Retina is the smallest among those products using a-Si. Sharp failed to solve the chronic problem of IGZO, i.e. uniformity, and its yield went down.
Notably, the burn-in is not visible to ordinary users; however, it doesn't meet Apple's standards. The company is trying to find a solution which may include bringing Samsung Display onboard as a supplier. Earlier negotiations with Samsung reportedly fell through; however, since it was the first to solve the problem, negotiations resumed.
Another solution may be to upgrade the display to LTPS.
As LTPS has more masks than a-Si, it costs more to produce, and the production capacity is smaller too. But stable supply will be possible, which is a definite plus.
Finally, Apple could relax its standards as it's not visible to untrained eyes; however, this may compromise the company's image.