I like your latest experiment, the effect is awesome. Did you use a picture for the radial light instead of making it default material, and did you include it in the influence- checking to see if my understanding is correct before I give it a try.
Your question, which I have indeed missed to answer. I apologise.
Objects: 100% transparent, 153 refraction.
Render options: TA with TA scattering correction and boost light, max ray depth 6 and TIR 2.
Sky: no HDRI but global Ambient must be fully white. Never mind the other sky settings.
High-contrast Image: H77 of the Wallpapers was used, rendered with the Spherical Mapper to get it into the spherical projection. The rendered images was exported as 48-bit TIFF (no 48-bit Dithering!). This images was multiplied with itself and the result again multiplied with itself. The resulting HDRI was converted to a conventional 8/24-bit LDRI BMP.
There is an outer sphere with the high-contrast image in the Diffuse and Ambient slots and both at 100%. The sphere doesn’t cast shadows.
Radial light with the name AnyName (not Background) with the settings for obscure light: neither diffuse nor specular, TA optimised with Gel and no inclusion or exclusion. The radial is large but smaller than the outer sphere.
The gel is special: It uses the same high-contrast image as the outer sphere in the Ambient slot. Ambience is set to 0.2 and has a positive hyper texture to boost the light beyond the normal. It doesn’t cast shadows.