Changing crystal shape causes surprising behavior with Liljequist parhelion, casting doubts on its identity as a halo form. Look at the simulations above, larger image opens by clicking.
I have simulated (software: HaloPoint2.0) the anthelic side of the plate parhelic circle with three crystal shapes. On the left are all rays, on the right only the 3-5-6-7-3 rays that I have often referred as the main contributor to the Liljequist parhelion. But the shape of light stripe created by the 3-5-6-7-3 rays depends on the crystal shape. This means that the definition of the Liljequist parhelion becomes so murky that we might as well forget about it; 3-5-6-7-3 rays are just rays that light the parhelic circle, nothing more.
The other ray that I have mentioned of being responsible for the Liljequist parhelion is the 3-5-6-7 ray. Its product does not change with crystal shape, but it is probably too weak alone to produce any visible brightening on parhelic circle. I think it was me who originally brought up the Liljequist parhelion. I apologize for the mess. I guess we should let go of Liljequist, but it has been around for a while, so that may be easier said than done.